Do ideal gasses really not interact with each other at all? I'm having trouble understanding how two different ideal gasses occupying the same space affect/interact with eachother. Presumably, the answer to this question is as in the title: Ideal gasses do not interact with eachother at all. 
However, this seems incredibly odd to me. I'll try to illustrate my confusion through an example.
Imagine a container divided into two compartments by a membrane passable by particles of ideal gas type A but impassable by particles of ideal gas type B.
Now suppose the container is filled with gas A. Clearly the gas does not mind the membrane and will distribute uniformly across the entire container. Now suppose we inject gas B into the left compartment, where it shall remain. Will the distribution of gas A across the container remain uniform?
On the one hand, if the gasses do not interact at all then gas A "doesn't know" about the injection and will remain evenly distributed. On the other hand, the total pressure in the left compartment will obviously be higher than that in the right one. Does that truly not affect the distribution of gas particles at all in this scenario?
Another confusing thing about this scenario is that we can inject gas B which is at a higher temperature than that of gas A already in the container. Clearly the two gasses will reach thermal equilibrium and the temperature across both gasses in the entire container will eventuallly be uniform, but how is that possible without interaction between the gas particles?
 A: Ideal gases are a model. That model intentionally leaves out some parts of reality. If you're asking me whether the model predicts that the two gases will indeed remain forever separate in their physical properties, my gut instinct is yes. With that said, of course the model is a bad model for any of those sorts of physical processes, precisely because it predicts this.
Semipermeable membranes like the one that you're talking about are actually a staple of biophysics, where frequently water can flow across both sides but a salt or protein cannot. These membranes feel an osmotic pressure to go one way or the other, based on the fact that the salt is pushing on them from the one way, and the water cannot provide a similar force from the other direction. So that aspect of the physics is very well known.
So if you are looking for more material to Google, I would start with “osmosis,” and also the phrase “partial pressure.” What you are describing in fact is the straightforward situation where gas A has one partial pressure, and gas B has a different one.
A: The gasses do interact in that they collide with each other with elastic collisions. 
In the first case, gas A fills the container which has uniform temperature and pressure. The addition of gas B in one section will raise the press of that section, the temperature remains constant. 
That is all that happens, you just have a container with different pressures on either side. 
A: The assumption of the ideal gas model is that there is interaction between the particles - the perfect elastic collisions between them.
The main consequence of the assumption is that the equation $PV=nRT$ holds, which means that the pressure of each of your containers' halves is given only by the number of particles, and their temperature (which corresponds to their average kinetic energy).
Thanks to the zeroth law of thermodynamics we know that if we have a spontaneous transfer of energy between two objects, Eventually they'll reach a state of mutual thermal equilibrium. in that case, the two gases (low temp and high temp) are transferring energy with elastic collisions, so you know they'll reach the same temperature.
As others here said before, there will be a different pressure in each half of the container. If you find this unintuitive, think of a single particle - the particles in both sides have the same average energy, so the collisions he'll suffer "will feel the same". The main difference is that in the side with the bigger pressure he'll have more collisions per time.
Intuitively one might say that a particle that goes from the low pressure half to the high pressure half would experience more resistance (than a particle doing the opposite way) as he's more likely to be "blocked" by collisions. In the ideal gas model we can neglect this kind of effects because we assume each particle covers a long distance between each collision. That's actually the reason that the ideal gas model doesn't work very well when the pressure/temperature ratio is relatively high.
