Don't focus on that other house... see the note in MLS listings "information deemed accurate but not guaranteed". The agent may not know the permit state of a room as a bedroom, but as long as it looks like it could be one they will probably list it as one. There may not have been any permits, or if there were the room may not have been considered a bedroom at the time the permit was issued.

You would have to consult your town or county's code for specifics. You can most often find these online. I would be surprised if you didn't have to pull a permit, since typically (at least in the bay area) anything more elaborate than painting requires a permit (and some jurisdictions require a approval even for painting). If you're adding framing you may be adding electrical too. These are things that may warrant inspection. Whether you decide to do this or just do what so many other people do and do it off the record is another matter. If it were a simple job of a non-bearing partition wall I might consider this, especially if I knew what I was doing or had a good contractor.

I am in the bay area also and agree with caben. Do not go by what the other house was listed/sold as. In pour plan we had a room with a skylight shown as a "prayer room". This did not get approval as it did not have window. From the city perspective you need a window to be a habitual place. So we changed the label in the room as "closet " and the same plan passed.
We still plan to use it like we want it but from the city's perspective it is a closet.

The city I am in is pretty strict and requires permits for most stuff.

Caben and Akshar's mom, thank you very much for your very helpful responses.