We have a happy screamer too right now. I have be advised from other moms who have gone through it, it is a phase and will too pass. I can't wait though.
We have had success with redirecting such as "do singing" and get him to sing a song (which because he is 13 months means "ahhhh" in his pretty singing voice). We basically try to give him ideas of other outlets to use his voice, since his intent behind the behavior is to hear his lovely loud voice. We also ask him to do other noises he makes like "do scary", "do barking", whatever we know is in his repetoire. We also simple say in a calm quiet voice every single time "no screaming". He understands what we are saying because when we say the word screaming now (if he is not screaming) he does a nice short loud scream and smiles at us (letting us know he has such a great understanding!!).
It is totally embarrassing in public and some days he screams more than others, but we are just glad they are screams of happiness rather than tantruming screams. We also do baby sign language though, so his screams are not means of communicating anything to us other than loving to hear his voice.
Good luck, I can't wait to read some other ideas that people have come up with. Maybe it will give us more ways to redirect his loud voice :)