Can I switch from an ATT iPhone 4S to a used Verizon 4S?

My mom is on Verizon and I'm on ATT. She recently got an iPhone 5S and she said she can give me her old 4S (I cracked the screen on mine). Can we do this? If so, can we do it from home or would we have to go to an Apple or ATT store?

Verizon would need to be used on Verizon with a 4S. If your contract is up on ATT then you could move to Verizon of course, but in that case I would do a new 2 year contract and get a new 4S free. Don't ever throw out your 4S though as you can trade that in even if damaged and upgrade, they will still give you some $ back for it.

No. Even if you were successful in jailbreaking the phone, the technology itself is incompatible. AT&T and TMobile use GSM (The system where you have a SIM card in the phone). Verizon and Sprint use CDMA (phones that don't have SIM cards). GSM and CDMA are two different technologies and a phone designed for one technology can't operate on a network that uses the other technology. An iPhone 4s supports either GSM or CDMA but not both