The three CCtCAP competitors, Dream Chaser, CST-100, and Dragon are all designed to carry a crew of 7.
Operationally, NASA plans on launching crews of 4, since that allows the station to support a crew of 7 (3 by Soyuz, 4 by US side) and the station has 'room' (4 sleeping berths in Unity, 3 in Zvezda, though that might only be 2) and life support for a long term crew of 7 total.
Direct handoff of crews does not seem to be likely, where, say a Dragon is docked to PMA-2 on the forward end of Node 2, and a second vehicle say a CST-100 docks to the zenith (space facing) PMA-3 and there is a Soyuz on the Russian segment. That would imply 7 on station + 7 more, but that might stress the life support system too much. (Shuttle brought its own fairly robust life support system).
More likely is indirect handoff, where a Dream Chaser, docked to PMA-2, undocks with a complement of 4, reenters, and a week later a CST-100 launches to deliver a crew of 4. Which is how the Soyuz handoff has been happening for most of the stations lifetime.
(Why indirect? For one thing NASA requires a spare docking port be available, so if they arrive, and PMA-2 is busted, they need to have PMA-3 available, and at the moment, there are only going to be 2 PMA's available for docking. For berthing (HTV, Dragon V1, and Cygnus) the nadir (earth facing) CBM port on Node 2, and on Node 1 will be left available for that, once some modules get minorly shuffled around the station. Right now, the backup CBM port is the zenith (space facing) CBM port, that is due to be replaced with PMA-3).