Confusingly, this is one of the words that we use for both parties in an asymmetrical relationship (cf. "namesake"): Both the person confessing and the person hearing the confession can be called a confessor. Per Oxford Dictionaries, the term is probably applied more often to the person hearing the confession:
- A priest who hears confessions and gives absolution and spiritual counsel.
. . .
1.1 A person to whom another confides personal problems.
. . .
- A person who makes a confession.
Some of the example sentences under definition 1.1 suggest a couple of possible solutions to the ambiguity:
‘Confession in the classroom takes many forms; therefore, the identities of the confessor and confessee are not always the same.’
. . .
‘How do we understand, not what is said between the confessor and confessant, but the dynamic that is produced between them?’
. . .
‘Sometimes confessing is better for the confessor than the ‘confessee’ and just makes unnecessary trouble.’
I note that "confessee" isn't in the ODO and still seems a bit ambiguous to me (does that last example really mean what ODO thinks it means?), but "confessant" is in the dictionary and seems more straightforward (if "fancier").
Also, in practice confessor is very often used with a possessive determiner, i.e. "my confessor" or "the penitent's confessor", which generally helps to clarify who is confessing and who is hearing the confession.
One more possibility, if the person is trusted to keep the confession private: Confidant works pretty well for confessions to people like close friends, personal advisors, and spouses. It works less well for a formal relationship like a therapist or priest, and doesn't really make sense for someone like a police officer.