I had a Verizon Note 4 and left Verizon for good after some hassles with that company and cost me a lot of money. I brought it to both AT&T and T-Mobile. T-Mobile which is suppose to be fast in the area, I couldn't even get 2mbps download speeds in the Store! I didn't even leave the store, just cancelled right then and there. That's ridiculously slow.
Then I brought to AT&T for a few weeks, I actually had a Verizon Note 2 on AT&T where everything worked fine I got very fast speeds, but ended up breaking my Note 2 screen, so just went to ATT and bought a Note 3 from them. When Note 4 came out, It was too expensive at AT&T, so went back to Verizon and got the Note 4. After leaving Verizon though, I had problems with the Note 4 never holding a consistent signal (h,hspa+, LTE), I could do everything except mobile hotspot, so it was using too much battery not holding a strong Signal. AT&T said they could not guarantee good service because of it Being a Verizon Note 4. So I just traded it in at Best Buy and paid for the AT&T Note 4 which I have now. So I read of people having no problems, but I had problems with the Verizon N4 on other carriers. I think it's better to go with the carrier specific Note of your choice with your Carrier. Just my experience with Trying.
It amazes me though how my Note 2, which was an earlier phone, worked better than the note 4 which I would think with Technology would be more adaptable between carriers. There's always a trade off somewhere when you try on another carrier. I didn't like losing the Hotspot, because I have tablets that are wi fi only. The speeds on AT&T were not that great either with my Verizon Note 4, but they weren't great on Verizon either, which was a great disappointment. The other day on AT&T I was getting 53mbps download and over 8mbps upload on 4g. I never saw above 28mps download and maybe about the same on upload on Verizon with my Note 4. I'm liking AT&T much better as far as that's concerned!