I think there are two distinct cases to discuss: the case in the movie; the case about the movie.
The case in the movie: it is 1964 and in a Catholic school in Bronx a conflict erupts between the principal (Meryl Streep) and the priest (Philip Seymour Hoffman). The principal is Sister Aloysius, a nun very strict about the discipline. The priest, Father Flynn, is just the opposite, a very natural guy, open to people and to the world. The nun suspects him of pedophilia. The suspicion will never be confirmed, never thrown away. Eventually it's up to us to decide, and our role of spectators is played in the movie by Sister James (Amy Adams), a younger nun who is trying to understand what really is, oscillating between the two. By the way, genial idea of using male names for the Catholic nuns, to stress out the strictness of their rules.
The case about the movie: the epoch Doubt was made and the epoch the picture takes place are very different. The sixties were the years of Vatican II; the Catholic Church was opening largely its windows. It was the conflict (often brutal) between new and tradition, between progressives and conservatives. This was then. Today the Church is facing the scandals of pedophilia (and the way they are handled by the Catholic hierarchy).
So, if we take the epoch of the sixties, we take the side of Father Flynn, a man open to modernity, empathizing with the youth, with their questions and their way of seeing the world, speaking the language of his epoch, a wonderful man suspected by a retrograde nun.
Only the movie is made today, for today's viewers, and we are focused on today's issues. So here is the question: once the nun had suspicions that the priest was a pedophile, what was the right way to take? To not follow a case without positive evidence? Or, by the contrary, to follow the case, to force him to come with proofs of his innocence? What was more important: his right to privacy or the safety of the boys? We can say that the movie leaves the case open. Nothing demonstrates positively that the priest is a pedophile; nothing demonstrates that he isn't.
Well, the movie brings something more: what if? What if the boy is born with another orientation and the priest is just understanding and protecting him? Maybe just because the priest has the same orientation? There is a key scene in the movie, the discussion between Sister Aloysius and the boy's mother (wonderfully played by Viola Davis), leading to an unexpected outcome.
And I think here is the doubt the movie is putting forward: more than the doubt of Sister James (is Father Flynn an abominable pedophile, beyond his openness?), more even than the doubt of Sister Aloysius (was she right in following a man without positive proofs?), there is the doubt of humanity. Human behavior is complex, each human case is unique and cannot be assimilated to a general pattern. Things aren't every time what they look like, we should always consider this question, what if?