My daughter is 11 and my son is 8. Can they legally share a bedroom?

Yes...

Possibly not, in public housing. Ask the main office.

Such laws are VERY local, often on the town maybe county level. They usually are guidelines set for public housing or by Family Services agencies.

However, an 11 yr old girl is going through or starting puberty. She really needs her privacy and sharing a room with a younger brother is not being fair to her.

First they should not have to,

Normally there is no "law" that says they can not. There may be of course rules in housing units .

Also if there is complaints CPS may have their own rules they will want inforced.

I understand that people are out of work, and often we have to live and accept standards we don't normally do.