I'm not sure that there is any 'legal' document saying which pilot should do what. There is probably plenty of advice or suggestions on what each pilot should do, but your company SOP or QRH should have a basic procedure somewhere. Of course, one should always follow the RA and ignore conflicting ATC instructions.
Having used TCAS in both military and civilian aircraft the procedures seem to be very similar. A healthy dose of common sense also applies.
TA:
PF - Try to identify the conflicting aircraft visually and call it out. Be ready for a possible RA.
PNF - Try to identify the conflicting aircraft visually and call it out. Be ready to respond if an RA occurs and the PF doesn't react.
RA:
PF - Disconnect autopilot and comply with TCAS RA climb or descent. (Be ready for a change of direction of RA). Try to identify the conflicting aircraft visually and call it out.
PNF - Advise ATC e.g. "Callsign TCAS climb/descent". Try to identify the conflicting aircraft visually and call it out. Monitor the PFs flying. Be ready to respond if the PF doesn't react to the RA.
(Warning- a TCAS "Monitor Vertical Speed" RA does not require a manouevre.)
Remember, the aim is to prevent a collison. Once you have returned to your assigned flight level, and told ATC that you have done so, and the heartbeats have slowed down, then you can 'discuss' who should have done what on a perfect day. As long as it all gets done!
Then, of course there is the RA in the landing configuration ................ or the fact that you should ignore a "Descend" RA below 1000ft .................. surely you must have this stuff somewhere in your company manuals?