It is perfectly possible for one aircraft to get only a TA while the other gets an RA. A common mechanism by which this occurs is the operation of the horizontal Miss Distance Filter in TCAS version 7.
If TCAS can reliably determine that there will be significant lateral separation then any RA that would otherwise be generated is suppressed. However if the the TCAS surveillance happens to be "noisy" then the determination is not reliable and so the RA is issued.
It is often the case that the MDF on one aircraft's TCAS suppresses an RA on that aircraft while the MDF on the other aircraft's TCAS does not supopress the RA on that aircraft.