Hi,
I am currently studying a Masters, which I will finish by January, 2015. But I've seen that the deadline for applications for funded PhDs starting in October 2014 is in January/February 2014.

So the question is, should I apply during this period (now)? Can I start in January 2015 if I got admitted? Or it might be better to contact the professor I would like to work with? Outside the application period, is there another way of funding my research project? I mean, even if you don't get a scholarship now, the potential supervisors themselves can fund your project?

Sometimes you can start earlier/later than the advertised date but it depends on the professor and institution. Often international, self-funded students start at random times but really universities prefer you to start at the beginning of the academic year.

Your best bet is to apply for Oct 2014 entry and then ask at the interview whether you can start in Jan 2015. I guess it will depend how much they want you as to whether they will let you do this or not. If they have other good candidates that can start on time they may select them over you.

If you start one without funding you are very unlikely to get funding later on.

Bear in mind PhDs are advertised throughout the year but most of those with funding will want you to start in October. Maybe you could delay starting one for a year and get a random job in the meantime?

But for instance, if the potential supervisor is funded by an ERC grant and is looking for a graduate student, that means the starting date could be whenever, isn't it?

Delaying the starting date for one year is not an option I'm considering right now... is there any chance I could do my master's project (between October 2014 and January 2015) with a potential supervisor and then continue doing the phd?

Technically the starting date is whenever, but most academics prefer their students to start at the beginning at the academic year because it's easier for administration, funding, welcoming new students, seminars, progress assessments etc.

I thought you were already doing your masters? If you mean can you take your current project to a new supervisor and finish the last few months and then do the PhD with them, probably not, no. The supervisor providing the funding is going to want you to work on their project. They probably won't take you on if you haven't finished the masters.