Verizon and Sprint are CDMA carriers and run on compatible CDMA frequencies and technologies (1xRTT, EvDO) and both their branded phones can roam on/off each other's network for both voice and data.
Having said that, the fact that they're primarily CDMA carriers means that you cannot just bring a phone onto their respective networks. The ESN/MEID of the phone has to already be in their database for them to activate.
So, taking a Sprint LG G2 and walking into a Verizon store or calling Verizon to activate it will get you a big fat NO, and vice versa.
Is it technically possible? Of course.
Worth the effort/hassle? No.
T-Mobile is a GSM carrier, no CDMA, so that's out of the question, same with AT&T, or any GSM variant of the G2.
Get a Verizon LG G2. Just keep looking carefully and you'll find a good deal.
I got my Sprint LG G2 mint in box for 325, clean ESN, no funny business. Love it to bits.
Epic Touch (Galaxy)
Nexus S 4G
Epic 4G Touch (Galaxy S2)
Galaxy Note 2
Moto X (Current)