We traveled with 2 beagles for the first year and didn't have any problems but I did notice some of the RV Parks had a 2 dog limit, so that could be an issue.
As for boondocking out west in the desert, you'll want to watch them as coyotes have been known to snatch small dogs that get left unattended. But overall out there is more space out west to roam and boondock so easier to find places away from others.

I agree with Ray, we have two small dogs lasha/datsun mix they travel real good for us. most if not all RV parks have on lease policy in park. most have dog parks where they can be off lease. as Ray said some have two dog limits but those that don't may charge you extra for having more than two. You can always call ahead and find out the rv parks pet policy for the park your staying at.
Also make sure all there shots are up to date. we take a copy of both our dogs vet records with us incase we need to see a vet on our travels. when crossing Borders i believe you need there record anyways. there are web sites that will tell you what you need, or someone on here that has traveled across the borders can tell you.
all in all you shouldn't have any problem.

You shouldn't have any problems out east, but out in the mountain states, mountain lions have been taking dogs as well. The most recent was just down the road from where we lived in Idaho.

Additionally, mind your dogs around the rivers, lakes and ponds in Florida, Alligators too have taken pets.

Good morning Catie! As a life long dog owner and lover, I respect owners who have control of their dogs, both physical control and barking control. That being said, unfortunately there are dog owners out there who think their barking dog is cute and do nothing to deter barking. And there are the dog owners who allow their dogs to wander, off-leash, in the camping area or on the trail.

If you want to be a good camping neighbor, don’t be that kind of dog owner. Be like Ray. His beagles were always on-leash and respectfully quiet.