With villas, it really depends on the landlord I guess. I live in a small compound with my wife and two dogs (one's pretty large at 42kg and the other is a 3 month puppy) and our neighbors also have a dog. We usually take them out on walks behind the compound where there's a big empty lot. However, you can't really walk your dogs everywhere (ie Corniche).

There's a beach in an area called Al Wakra (which is about a 10-15min drive from Doha) where my wife and I usually take our two dogs out for a run and swim on Friday mornings with another friend who also has a dog.

You'll see quite a few dog owners out there on the beach with their pets and as long as your dog is well behaved it's all good. It's a pretty big public beach and not crowded at all so you're bound to find a huge "vacant lot" where you can do your own thing and not bother anyone.