"Non sai se ha un cane."

[HavardF]

Would an Italian say that sentence without any pronouns?

[jesslc]

Probably.

It's highly unlikely that this sentence would be used in isolation - so you would be able to tell who the subject of the verb 'ha' is by the context (ie. by the earlier part of the conversation or by gestures/body language). If the subject can be deduced from the context, pronouns are not used - unless you specifically want to place the emphasis of that sentence on the subject.

[Muttley_]

Since the verb carries the person and number, subjects are often left out in Italian. In facts this sentence sounds natural.

[balazs81]

yes, or the english translation should be different