This person would be called private.
"Private" is used to describe not only information, but also people who don't share personal information freely.
There are a variety of expressions to describe someone who does not wish to disclose other types of information, such as "secretive," "tight-lipped," and "discreet." However, "private" seems the best for the situation you've described.
Two other possible candidates are unforthcoming and guarded.
From Oxford Dictionaries online:
private
1.3 (Of a person) choosing not to reveal their thoughts and feelings to others: he was a very private man
unforthcoming
1 (Of a person) not willing to divulge information: the sergeant seemed unforthcoming, so he enquired at the gate
guarded
Cautious and having possible reservations: he has given a guarded welcome to the idea
Best Answer
Aonanist. An onanist being a man who masturbates or practices coitus interruptus, an aonanist would be someone who doesn't (similar to social/asocial).
I don't think onanophobe is right because that would indicate a fear of onanism while there are certainly other reasons for not practicing it.