I don't think you can do that since T-mo uses GSM and Verizon uses CDMA.
Actually, the Verizon S3 and Note 2 have had GSM radios. The Verizon S4 will as well... and even have LTE on T-mo's frequency (band 4 -- AWS), since that's Verizon's next LTE rollout.
What it will *not* have, as far as I know, is GSM/HSPA on AWS. So you won't get T-mo 3G unless you're in an area where they've refarmed 1900mhz to 3G (which is actually a lot of them now, and should be all of them in the future). If you are, though, you're good to go for both 3G and LTE.
The only catch is that you may have to root the device to edit the APNs for T-mo use.