How can I best divide a connected two room into separate bedrooms?
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.