Dogs that urinate in the manner described generally are expressing a behavior called submissive urination.
Submissive urination is a form of communication. In the dog world, when a submissive individual interacts with a dominant individual, the submissive dog often releases a small bit of urine. This is a sign of submission — it’s your dog’s way of saying that she is harmless and doesn’t want any trouble.
Sadly, when dogs engage in submissive urination in front of humans, trouble is exactly what they get. Always resist the urge to scold a dog after submissive urination — scolding or punishing will only make the problem worse.
The best way to deal with submissive urination is to behave in a low-key fashion whenever a triggering circumstance arises. For instance, a submissive dog may urinate when a new person enters the room and greets her. The simplest way to prevent urination in that case is for people to ignore the dog for the first few minutes after they arrive.
Also, avoid getting the dog really riled up. You know what activities tend to cause urination, so avoid those activities until she outgrows the problem.
And, speaking of outgrowing the problem, there is good news: the overwhelming majority of dogs outgrow submissive urination by the time they’re two. You should not need to avoid excitement and warm greetings indefinitely.
Finally, it is wise to schedule a veterinary exam any time a pet engages in house soiling. Some medical conditions, such as bladder infections, can trigger behavior that looks like submissive urination.