You just need to update your APN settings. The APN settings are defaulted to the Carrier, Verizon in this case. If you put in someone elses sim card you need to adjust the network settings as well. There is an automated way of doing it at the website unlockit.co.nz, they have preconfigured apn profiles for all the carriers, works like a charm. I actually put my TMobile sim in my Verizon iPad 4 this morning and no problems at all.
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Follow up, Sims are not matched to device, they are portable and switchable. Some devices however are LOCKED to only accept the same carriers sims (like AT&T is), but Verizon ships their iPads unlocked so they work with any carriers sims.