I really doubt that dispatch would monitor their own base frequency. Having your own voice come back in your ear is slightly annoying, but if there is any delay (as any modern console has along with the rest of the infrastructure) it would be quite unsettling.
The CHP used to have the ability to switch from base to car-car kinda "on the fly". Do they still have that or just change channels now like we all do? The Tesla recording is inconclusive as to whether they were on base or car-car and just switching back and forth.
This is a SWAG on my part if CHP is using this, but most consoles can do both acoustic and RF-based cross muting - that is to say, if the dispatcher keys up resource X, the audio from resource Y is muted for the duration of their transmission. CHP typically does have base frequency RX remote bases for given channels at some of their sites, which is how they would hear the car to car traffic.
My recollection from the last time I saw the Rockwell-Collins CPVE setup was they had a custom mic with two switches on it, similar to the old GE radios. Of course, if they're on their extender, they need to switch channels on the portable to gell the car to TX on the base freq, as opposed to the mobile freq.