Understanding the Practical Application of a Simple Antenna made from Coax
What these two do, whether they just form a dipole together, or are two separate monopoles (with bad ground) is impossible to tell based on the info we have.

I find it less likely they form a dipole.

Maybe there's just one receive and one transmit antenna, to save on a directional coupler or antenna switch.

Maybe these antennas are there for diversity reasons – for example, to decrease the likelihood that all antennas you have are subject to fading, you just use more antennas. \$P(\text{all antennas faded}) &lt; P(\text{at least one antenna is OK})\$


  
  In what way does the shielding effect the antenna?
  


I'll nitpick on your terminology for a second, because it makes a difference later:

That's not primarily a "shield", that's the outer conductor of the coax cable, and just as important as the center conductor at transporting energy. (In fact, you can have a hollow waveguide without the center conductor.)

Together with the center conductor and the dielectric material, it forms a waveguide.


  Specifically, if you were to cut the wire down (significantly changing the overall length of wire) but strip away shielding such that the exposed length of center conductor is unchanged, how would this effect the performance of the antenna?


Impossible to tell. The coax might be somewhere between impedance of the trace on the PCB and the monopole feedpoint impedance, and act as e.g. a \$\frac\lambda4\$ impedance transformer. In that case, the length of the isolated part is critical.

Maybe it's just a best-effort matched transmission line, and changing the length of it doesn't affect the impedance. 

But: in my theory (two separate monopoles), these antennas are separate for a reason (typically, diversity to increase robustness or speed of transmission); making their leads shorter would put the antennas closer together, which would have negative effects on channel independence, and hence, on performance.