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In Ohio, the state I live in, it's only legal if you're riding the horse. This is because a horse and buggy is subject to all of the laws associated with motor vehicles, meaning that they need turn signals and lights. That also means you cannot legally operate a horse and buggy while distracted or under the influence.

If you're riding a horse, you're only subject to Ohio Administrative Code 1501:46-13-10, which doesn't forbid you from riding drunk, nor does it forbid you from riding while checking your cell phone, but does forbid you from riding too fast, in the wrong areas (such as a highway), or from hitching your animal to anything that it might damage. These regulations apply to all saddle animals, including donkeys and elephants.

I have to assume the regulations are different where you live, though. We only need to regularly update our traffic regulations to be amenable to horse and buggies because we have a lot of Amish here, and so horse and buggies are actually pretty common, but I assume if you live in like Colorado or somewhere outside of the American corn belt you don't have a lot of Amish and therefore the laws are likely to be different.