Renting an incredible apartment with two other friends, catch is that it is 2-3 bedrooms 'depending on disposition'. The room/s in question are shown in the photo. What's nice is that each would have a door to the hallway in the event that they are closed off. I'm just looking for recommendation on things to put in between for privacy and sound insulation (if possible)
If you are considering separating the alcove in the first picture into a separate room, you may have a problem with building codes, and more importantly, safety. Even if the alcove has access to the hallway, it does not appear to have a window, which may be considered a hazard, both as to air and alternate egress in the case of a fire or other emergency.
You need to check with the housing or building authority before separating space.
You also may have a problem with your landlord, depending on what your lease says. Usually renters are not permitted to erect anything structural without permission.
If you are merely asking about a curtain for the archway, there are sound insulating curtains you can find online, but they are likely to reduce noise, not eliminate it.