Mostly old information. Most smart phones have the right frequencies to work in Europe now. Ten years ago, before most people had smart phones, this was NOT true of flip phones.
Understand: Europe uses GSM technology for mobile phones. In the US, T-Mobile and AT&T also use GSM, but they use different frequencies here in the US than are used in Europe. Older phones that worked on T-Mobile and AT&T in the US - flip phones - may not have been "quad band" phones that had the right frequencies to work in Europe, too. As late as 2009, my T-Mobile Sony flip phone did not/was not compatible with Europe.
Verizon and Sprint used to use exclusively CDMA technology, which is completely incompatible with GSM. So an old Verizon or Sprint flip phone (no SIM card like a GSM phone) would not work in Europe - period. Newer phones use LTE which is a bit different.
But most newer smart phones that work on Verizon and Sprint can roam on GSM now and have the right European frequencies to work at least on slow data networks and for voice calls. Otherwise, Verizon and Sprint wouldn't be able to offer international roaming (for which Verizon charges a lot). And most T-Mobile/AT&T GSM-compatible phones have the right European frequencies, now too. Note that they may only work on SLOW (2G or edge) data networks in Europe, though. It entirely depends on the phone's make and model - nothing to do with the carrier itself, actually. I have an older Moto E phone designed for Verizon that can work on 4GLTE networks in the US (even on T-Mobile/AT&T) but in Europe it can do only slow 2G networks - barely usable for data but OK for calls.
Check the website frequencycheck.com for your phone's exact make/model to see if it is compatible with European mobile networks. Note that many phone models come in variations with the same model name; there are numerous versions of the Moto E for example (I have several including the Indian version that works great in Europe but not so well in the US).