As I remember saying "live by us" and "live with us" is equal in meaning in English.
It means essentially the same. "bei uns" refers a little more to the place, while "mit uns" refers a little more to the fact, that you're living together (If you're pedantic, it doesn't necessarily imply, that you're living at a certain place, you could also be moving aroung or something).
"mit" = "with", "bei"="near (in the same house/flat)"
However in most cases it sounds more natural to use "bei uns".