There are a few cases.
First case, you measure your particle then you write a letter to your friend and tell your result to your friend before your friend measures their stuff. They can be amazed to know the result of their measurements before they do them. Or they can be unamazed since by then your measurement has had time to affect them without violating relativity. Nothing interesting.
Second case, same thing except you get a letter from your friend before you measure your stuff. Again, nothing exciting.
Third case, like the first case above but you don't bother sending the letter, but you could have, so nothing interesting.
Fourth case, like the second case above but your friend didn't bother sending a letter, but they could have, so nothing interesting.
Fifth case, both of you measure your stuff before you could have sent a message. Is this spooky? Well, neither of you caused the specific result you got. You could think your result caused your friend's result and someone else could think your friend's result caused your result. But neither of you could control what you got so you couldn't actually use what you did to affect them. You aren't affecting them, so no violation of relativity. You are both getting results and neither of you controls the results it's like someone else is giving you both results that happen to be correlated. This isn't very different than you and your cousin getting a holiday checks from your grandma for the same amount, it's something that happened to you both and it's something that is correlated, but you aren't affecting each other by getting the same things.
The distant people aren't affecting each other. So no serious violation of relativity. For case five, you might ask since who measured first is frame dependent whether I am saying we can't tell who affected whom. But they don't even have separate states that they can affect.
If you write down the Schrödinger equation for a pair of entangled particles and a pair of measuring devices and evolve it according to the actual Hamiltonian for the actual setup (an equation few people ever setup in their entire life, even amongst physicists) then it is clear what a measurement does. It takes a beam in configuration space and bifurcates it into separate beams while possibly polarizing the separate beams.
The bifurcation happens in the part where the measurement happens. But the beam itself exists in configuration space and always will.
It might help to be very very specific about this, so let's pick a concrete example, you are going to measure spin with a Stern-Gerlach device, the beam to measure the spin will travel in the y direction, have some thickness in the x direction and split into two beams that in the x direction do not overlap. In one dimension you could track he x with of the beam over time as starting out as a single line say from 2-4 and over time it splits like a tree branch into two separate lines one say on the left from 1-2 and another branch from 4-5. If you had time go in the y direction it does look like a branch, of a fork in a river. But the actual beam exists in confirmation space (that's normal for quantum mechanics if people don't tell you thus they are oversimplifying but you need it in this case), which has x, y , and z for each particle so we need to have beams that have two x coordinates one for each particle.
So imagine you have a blob that has each x be in the range 2-4 you could draw it as a square in the xy plane with the x being the x position of particle one and the y being the x position of particle two. What a measurement of particle one does is bifurcate the beam so a single beam with spread of x from 2-4 continuously evolves into two beams. It becomes a beam from 1-2 and another from 4-5. So that blob in the xy plane splits down a vertical line down the middle and becomes two rectangles that move apart to have a rectangle of emptiness between them by having each piece move in the horizontal direction. That is what a measurement of particle one does it splits blobs in configuration space with vertical lines and separates them by moving them horizontally.
What does a measurement of particle two do? It splits blobs by making a horizontal line down the middle and then separating them by having them each move differently in the vertical direction.
And that is what they each do. Its like cutting a square pizza, you van slice it one way first or the other way first but it isn't really affecting each other. Because we know what each does, it separates every piece according to the rules. Measurements of particle one make horizontal lines and move them apart by moving them vertically because we are using the vertical direction to represent the locations of particle one and that is what we are measuring. We don't really care if the beam is already separated in another direction.
If you do both at the same time it just means your pizza rips into fours pieces at once instead of ripping into two and then each of those ripping into two. You aren't really changing how the other one operates. There is a wrinkle I'll get to later about entanglement, right now I'm just addressing mutliparticld measurements
But first there is another important detail for a single particle measurement. For particles with spin, there is a spin state as well as a spatial state. When you measure the spin, the location of that vertical or horizontal line depends on how the spin state compare to the orientation of the Stern-Gerlach device, and by the time the beams are separate each (now separate) beam has that particle's spin state polarized to one of its eigenvalues for that orientation's operator.
The Schrödinger equation is unambiguous about how the wave function evolves, so if you have a device that sends spin up to the 4-5 part and sends spin down to the 1-2 part and the state has more spin up than spin down then the middle part of the 2-4 beam, the part nearest 3, actually deflects right, towards the 4-5 part.
But if you have a device that sends spin up to the 1-2 part and sends spin down to the 4-5 part and the state has more spin up than spin down then the middle part of the 2-4 beam, the part nearest 3, actually deflects left, towards the 1-2 part.
So the Schrödinger equation is utterly clear and unambiguous, but the actual detailed motion already depends on trivial kinds of details like how you calibrated your device.
The wave always separates the positions of something, maybe the locations of electrons in your photodetector, maybe the locations of electrons in your computer or in the ink arranged on your lab book. As long as something is changed based on the outcome of the experiment, then the beam separates, and this is normal.
And it is normal for the details to depend on the details of how you did it. But all you know when you measure one particle is whether it separated and then it can interact with other things based on those different places where it can be, it can interact with photodetectors and such in the 1-2 region or it can instead interact with photodetectors in the 4-5 region instead. Your stuff doesn't really act differently based on whether it separated in the direction corresponding to the other particle.
So that is my point the actual beam is like a series of rectangles in the actual configuration space and all you do is separate along the directions you have access to, and all the other people far away do is separate things along the direction they have access to.
So they aren't really affecting you. You separate by making vertical lines and moving things horizontally because the horizontal direction is your stuff. They separate by making horizontal lines and moving things vertically because the vertical direction is their stuff. You aren't affecting how or what they do.
And that's literally what the Schrödinger equation says.
And each of you can interact with your stuff and your stuff is only sensitive to where you particle is, so it interacts with how the beam has split in the horizontal direction and the other people's stuff interacts with how it split in the vertical direction.
So that is what I mean by you not affecting each other. But remember how I talked about the beam splitting causing the spin state for that particle to polarize in a way related to the orientation of the device? Now we can talk about entanglement of the spins. And the only unmentioned things will be the effective irreversibility of measurements and passing entanglement up the measurement chain.
When the spin states were entangled you didn't have single particle spin states, so when you polarize one, you polarize the other too. You create spin states for both particles.
So if you measured yours first you'd make a vertical line and separate them horizontally and the left one would have say spin up and the right one has spin down. But what do the other people see? The just see a beam that in the vertical direction has not been split and still has equal parts spin up and spin down.
Since they are separsted in the horizontal direction by that vertical line, they are technically now orthogonal. But guess what, different spin states were already orthogonal. So absolutely nothing has changed for the other people, they have a beam in the 2-4 range in their direction that also has orthogonally two different parts for two different spins. Nothing has changed and they know what to do when they measure, they split the beam with a horizontal line and separate them vertically. Where you draw the line depends on how spin you have if it is all the spin you are measuring it goes on one edge. So if they are entangled to have opposite spins you get just two rectangles one in the rectangle 1-2x4-5 the other in the 4-5x1-2 (assuming they oriented both their devices to send spin up to 4-5).
But each just sees two beams, one in 1-2 and one in 4-5. Each sees a full 50% of the area in each section of the beam (so sees the results 50% of the time if you choose to go the level of probability, we aren't and don't need to since I am merely talking about what the Schrödinger equation deterministically predicts for the actual experimental setup including the setup of the devices we use to measure them, which is why I keep mentioning whether the device sends spin up to 4-5 or sends spin up to 1-2 whereas at the probability level I could say you just get a result and ignore how it happened, but then people could argue, I'm sticking to just what the Schrödinger equation says). Each knows what a measurement does, it splits a beam horizontally/vertically based on how much spin is in the direction of the device and then interacts with the separated beam in an entirely local way based solely on how the beam separated in that horizontal/vertical fashion.
They simply don't change how the other one acts. The closest a change you see is whether your line separates a unified blob or whether the blob was already broken even though to you it looks like an unseparated beam, but you are still just pushing each section of the beam according to how the spin agrees with your device its just that the beam really always was in configuration space.
This is also actually why the double slit pattern can be destroyed you are separating the beam in that other-particle direction when you do a which-way measurement and so the beams just don't overlap to interfere anymore.
If this is at all confusing, then you might want to first see what the Schrödinger equation says for a single particle spin measurement when you describe the measurement device and process and track what the Schrödinger equation says about how the wave evolves and bifurcates the beam. The entanglement is throwing a level of complexity on top of that and most people don't learn that in detail.
So when I said you are not affecting the results I'm saying that when you separate horizontally the beam is still unseparated vertically and that their measurements and interactions still do the same things they always did.
There is a still an unseparated beam in your direction of separation. There is still 50% of the beam in the two different spin states. Your stuff still only interacts with your beam. And you never ever notice (or learn) that it did indeed spilt into two squares instead of four rectangles (like the Schrödinger equation predicted) until enough time has elapsed for the separation in one direction to start to effect the dynamics of the other, i.e. when enough time has passed to send a letter.
