Does AT&T use Verizon's towers?

I first moved into this home not yet a year ago and being in a rural area, have exhausted my search for wired service.  I first tried Verizon and it was blazing speeds for cellular but the plan was not what I wanted so switched to AT&T - with whom I am quite happy with.  However, the AT&T speeds are 1/4 or less than those of the Verizon service.

Thinking I'd improve my service, I purchased a directional cell antenna system.  It has helped but revealed to me that AT&T is not using the Verizon tower that's just a couple miles from me to the NW.  My service (best AT&T signal) is coming from a town 5 - 8 miles away in the southern direction according to tower searches and signal strength.  And it's a third party tower although I don't know if that even makes a difference.

Aren't the cell towers these days shared?  If they are, how does one check to see if a carrier has actually established usage of a tower?  Perhaps something else is going on and it's not that the source of cell data has changed but instead some landscaping has changed and the Verizon test initially was a fluke.

Thank you