Blacksheep,
I have to concur with 411A, for TCAS I you do not need a transponder at all, only the threat aircraft does, the TCAS I system can have its own transmitter/receiver independent of the transponder.
TCAS I will happily work even if the threat aircraft only has a Mode A transponder. TCAS I provides a flight deck display of traffic in the vicinity and, if appropriate, alerts the pilot that an aircraft presents a potential threat by generating a Traffic Advisory (TA).
TCAS II will provide TAs and, if the system calculates a risk of collision with an intruder aircraft, which is equipped with an altitude reporting transponder or TCAS, recommend avoidance maneuvers, in the vertical plane, to the pilot by generating a Resolution Advisory (RA). TCAS II you can still get an RA if the threat aircraft only has Mode C (for the altitude encoding). For a co-ordinated RA, both aircraft need Mode S for the inter aircraft data link.
TCAS II equipped aircraft use the Mode S transponder connected to the top and bottom antennas to do the transmitting and receiving.
