I agree with Alexander Kilcoyne, and I would add, that in spoken form linguistics could help to understand the topic or have a general idea. The longer you can listen, the easier.
If for example the character would live for a month in a druid cove, hearing the language nonstop, he might even be able to learn it a little.
On the other hand, using magic, it should be able to mind-read some teachings of a druid. Geas and other magic might also be possible.
To help keeping their language more secure, the druid might change a bit of their language every month, or year, or so. So the character would loose the language over time. This however only makes sense with spoken languages, where members see each other quite often.
So I would say that "secret languages" aren't 100% secret. With enough resources, you can learn them. At least that is what I would rule as a DM. Other people might judge that even divine favor grants the language secrecy, and that anyone who knows it, who isn't a druid will anger the god(s).
Learning druidic for a non-druid is kind of like learning Masonic secrets for a non-Mason: There's a big fuss made about secrecy and so on, but there've been enough people with loose lips over the centuries and a few who broke their vows and then wrote books such that anyone with a somewhat determined interest (but not enough to take a one level dip into druid) can probably find it out.
I expect there are a few private libraries with a copy of THOSE DERN DRUIDS KICKED ME OUT! I'LL SHOW THEM! HERE'S HOW TO SPEAK THEIR 'SEKRIT' LANGUAGE! The dialect is probably a few hundred years out of date, and would sound like someone who'd learned English from a copy of one of Shakespeare's plays and no pronunciation guide, but for eavesdropping on modern druids and figuring out what they're talking about? Sure, it should work.
That said, finding a copy of one of these books is likely difficult, as druids would be out to destroy them, and most public or semi-public libraries would not want to keep a copy in house lest pissy druids send an infestation of bookworms or just set fire to the place.
So, I'd say you'd need to find the local antiquarian book dealers and loremasters then make an extremely high DC Diplomacy check to find out if they know where such a book might be found, and they would also be making a Sense Motive check to see if you're a druid or a friend of a druid looking to destroy their library or just some chatty Cathy who'd be liable to blab to the druids too. Yes, you're a highly charismatic bard and your Diplomacy will make them like you a lot, but that's not the same thing as trusting you, and getting such a book? It's going to take a lot for "Inquiries" and "Asking my fellow scholars" and so on, even if the truth of the matter is just the antiquarian going upstairs, picking up Volume III of The Boring Edicts of King Blarg IV, slitting the binding, and taking out the little chapbook that druids would kill to destroy.
To help keeping their language more secure, the druid might change a bit of their language every month, or year, or so. So the character would loose the language over time. This however only makes sense with spoken languages, where members see each other quite often.
It could be so complex as to have different words during different seasons, and different contexts for the words based on the time of month.
It could be so complex as to have different words during different seasons, and different contexts for the words based on the time of month.
I was going to suggest the same thing but add in lunar cycles and possibly even environments. Druids who spend more time in the arctic might have 17 different words for snow.