What you ask is an open question for the Everettian (many worlds) interpretation of quantum mechanics.
In physics we really don't like to talk about something like "consciousness", and leave that, rather to philosophers and neuroscientists. This approach was fine for physicists prior to the discovery of quantum mechanics because it was possible to come up with a very simple naive model of consciousness. Namely that the state of our conscious mind (our subjective experience) is correlated 1:1 with the physical state of our body and brain. i.e. if you know the position and velocity of all the particles that make up our body and brain, and you know the electric and magnetic field everywhere you could, in principle, determine the corresponding mental state.
Whether this is a good or bad philosophy of the mind is a question I'll leave to philosophers. What matters is (a) is it is a fair idea worth considering and (b) my guess is that most physicists tacitly assume some model of the mind that works like this, even if it isn't something physicists regularly think about.
But quantum mechanics poses a problem to this naive dualism. In short, if a human performs a measurement on a quantum system that is in a superposition then, under the unitary evolution of the Schrodinger equation (i.e. quantum mechanics with no collapse, aka the Everett or many worlds interpretation) then quantum mechanics predicts the human will become entangled in with the particle in such a way that the matter making up their brain will be in a superposition of two distinct states. The question is then which state does the human experience if their brain is in a superposition/entangled state with having seen both spin up and spin down? This is the same as Schrodinger's cat, but more striking since it involves humans and conscious experience. This question is also essentially the measurement problem.
The Copenhagen response to this abominable theory was to say: There's no way a humans brain can be in a superposition (because we'd have no way of understanding the human's mental state in that case*) so the superposition must "collapse" if it "gets too big". The big problem with the Copenhagen interpretation is that it does not, in any scientific terms, specify under what conditions the universe follows the unitary Schrodinger equation and under what conditions it follows the non-unitary collapse dynamics.
The Everettian response to this situation is to say "well, let's just say there is no collapse postulate". In the Everett interpretation the math of quantum mechanics is as simple as possible. Only unitary evolution of the universal wavefunction, no collapse to worry about.
The major flaw with the Everettian interpretation is that, while physically and mathematically simple, it forces us to abandon our naive model of consciousness that tells us there is a simple 1:1 correspondence between the physical state of our brain and our conscious experience. The simplest forms of the Everett interpretation give no replacement theory of conscious experience and this is their downside as a solution to the quantum measurement problem.
There are some folks thinking about a solution to this problem, but it is quite fraught and I haven't seen anything convincing. So suffice it to say, the answer to the your question is no one knows the answer to your question, it is an open question in the foundations of quantum mechanics.
Some speculative possibilities:
- Maybe there are many subjective observers that correspond to what we would normally think of as "you", one for each mental state of the brain in the universal wavefunction. Unfortunately this idea can't get off the ground because the (1) "terms of the universal wavefunction" change if you change the basis you use to represent them (2) The number of "branches" of the wavefunction can't really be countable enumerated. It's more of a continuum and (3) what if 3/4 of your brain is in one particular state but the remaining 1/4 is in a superposition of 2 or more states? Are there two conscious observers in that case? what if it's only 1/10000 of your brain that is in a superposition? Do we still need two observers?
- Another possibility is that your physical state is in a superposition, but your mental state "collapses" into one possibility all the time. The approach also has problems. Does your conscious state persist in one "branch of the wavefunction" or does it flit between different branches from moment to moment (and you just don't realize because your memory (stored in your brain like data on a harddrive) flits along with your mental state. To me this approach (in contrast to the previous one) is at least philosophical tenenble even though it's hopelessly solipsistic.
With all of this said, the Everett interpretation is still my favorite interpretation, I just point out the above to clarify it's weakness, bearing in mind that ALL approaches to tackling the measurement problem today have some weakness. What I like about the Everett interpretation is that the physics is clear and simple. The problem is shifted onto the philosophy of mind rather than physics, and the hard problem of consciousness is another known open problem. So this apporach sort of combines the efforts of work on the hard problem of consciousness and the measurement problem. All of that said, the onus is on proponents of the Everett interpretation to generate and vet novel theories for the relationship between conscious experience and the entangled physical world**.
*This probably wasn't explicitly part of the justification for introducing the copenhagen interpretation, but in my opinion, it is the implicit motivation for the theory.
**Just like the onus is on proponents of the Copenhagen interpretation to give scientifically well-defined criteria for when the wavefunction evolves unitarily and when it evolves non-unitarily.