This is a notation site, and I am talking here about the situation with regard to the notation of a tune. It's often confused with the rights that apply to recordings, but that's a whole other issue, and it's not relevant. I'm also UK-based, and I'm talking about my understanding of the situation in the UK - though I think it also applies to most of Europe and the US. Don't trust me too much, I'm not a lawyer. If it's important, check with someone who is rather than believing this. But, it being relevant to this site, I have tried to get my head round it, and this is what I think you need to know :-
Copyright on a tune comes into existence at the same time as the tune itself, and belongs initially to the person that makes the tune. It is, however, property - intellectual property - and as such can be sold, or given away. It lasts for 70 years after the death of the person who made it, and can thus also be inherited. After that, a tune passes into the "public domain" and is no longer subject to copyright.
What this means is that, if you want to make use of a tune that's in copyright (to record it, say), you need to make a deal with the copyright holder. Traditionally, this involves giving them a bit of money, but it's up to them (like the "Free Software" licenses - if it's someone's property, they get to say on what terms other people can use it). Most countries have an organisation that acts as an intermediary, a middleman who can arrange the standard deal, take that money and pass it on to the right person (in the UK, this is the MCPS). If you have material that you'd like to put onto this site, you should also bear in mind that the same applies to publishing it - you need a copyright-holder's permission to put it up here for other people to see. In addition to this, transcribing a tune from its original source into the format used here is also work, on which the transcriber might want to claim rights (if anybody does, I'd advise them to say so somewhere in the notation, probably in the 'Transcriber' field).
Many people are happy to see their work out here on the 'net for others to enjoy. Some people aren't. Even those who'd be pleased to have it happen would very possibly (and reasonably) expect the courtesy of being asked. The only way to be sure is to do that, ask them.
The problematical bit is that in traditions where music is passed around by word of mouth, this can be a very uncertain business. If a tune's been passed around a lot it's not always easy to know for sure who made it (and, as above, the copyright-holder might not be the same as the person who made it. A record deal, for instance, might involve passing rights over to the record company) ... it can get messy, it's always possible to discover that you've trodden on someone's toes without realising it. There's no good answer to this, I'm afraid. In such cases, you just have to try and sort it out with them.
I guess lots of us have tunes we wouldn't want to make public (copyright's only one possible reason. I also, for example, have attempts at making my own tunes, which I don't quite want to throw away but am too embarrassed about to want to show to anyone else), and to be properly useful, the site needs to deal with this; which it does, you can keep things private. Please use this as appropriate.