This got me interested so I brushed off my notes from grad school--GPS predecessors like the US Navy's Timation first used quartz oscillators, then graduated to atomic clocks in orbit. However, even now ground station atomic clocks issue periodic time corrections to satellites in orbit. Since we can get accurate time of flight to Mars, I imagine satellites without atomic clocks could be used, albeit with less stable pseudo-range estimates. This could be overcome by using more satellites, giving more favorable pseudo-range variances.

Are those actually necessary?

Not an expert, but the whole network is on communication (with perfect understanding of the satellites locations no less), shouldn't they be able to run an ntpd like protocol to stay in sync, and just sync ground clocks to "starling consensus time" instead of "real time".