if you want to hear from both antennas simultaneously and separately, you first need a transceiver with two receivers built in and two antenna inputs - most modern high end transceivers have that or have an option for that.
Many ops put in switching for multiple beams to xmit on one or the other or both and with older transceivers with only one RX built-in, listening follows that but you don't get the L/R separation, you get both in both ears. To do what you want, you'd need more complex switching control to have the TX SO239 output of the xcvr going to the 2 antennas during a transmission, then when PTT drops have the second SO329 input (non xmit) on the xvcr open/disconnected and have receive paths from each antenna to the two xcvr SO239 connectors.
There are many off the shelf computer controlled antenna switches that do all that.
If you have enough separation between the two antennas, or they are of different polarization, you can also (with a modern xcvr with 2 receivers built in) use them simultaneously for what is called diversity reception. Often when QSB hits on antenna, the other is more at a peak - short fades get filled in. It is really noticeable when done right.
Most multi-op and Single Op 2 Radio contesters have some version of that running. Back in the old days (80s/90s) I operated at the W3LPL multi-multi and we (and other big MMs) had beverage antennas on each band with two xcvrs and one op would call CQ while the other op could tune through the CQing and find multipliers or when things were slow find unworked stations calling CQ. Now most SO2R single ops are doing all that, too with multiple antennas. If you notice a slow response from a really loud station calling CQ, often it is because they are finishing off a QSO they were listening to on radio A while radio B was calling CQ!
73 John K3TN