Some competitions are exploitive. Some are scams.

Any competition that is being run with the intention of generating "ideas" for a real project, or for a project that's really being considered, should compensate the competitors for their design work.  If there's no pay then the hosting organization is getting free work, and this type of competition perpetuates the general public's perception that architects and designers are clamoring to provide their work for nothing more than their love for their "art".  If you enter this type of competition you are part of perpetuating that problem. 

If a competition has an entry fee - particularly if it's a competition geared toward students or entry-level professionals - you should look into the hosting organization:  Are they a real organization with some history of activity other than running competitions? Have they held previous competitions? If so are the winning entries viewable somewhere online? And if you research the winners' names, do they seem to be real people with previous existences?  If they're real people, can you tell how many entries there were, and does it seem like anybody did not win?  There are a lot of competitions that are run solely in order to generate entry fees, and may not award more than a small fraction of fees collected, or in some cases do not award anything at all to any real person.  There are also a lot of  "competitions" that exist to dole out vanity awards to anybody and everybody who pays for them - which is not always apparent to all those who enter.  (These are like the architectural equivalent of those compilations of high school poetry that send a book and a "congratulations, you're published!" letter to every kid whose parent paid the "entry fee" to submit their work of genius that semester, or those "Who's Who" series of publications of "notable" people in their fields, which include everybody notable for having paid the nomination fee.) 

Competition is exploitative in its nature. First there are real project competition and fake ones. In the real project competition, especially ones with huge compensation and contract, requires pre-qualification selection, it is geared toward established design firms.Then there is real but open competition, the prize is usually the contract/money for winner, the project is often much smaller. Lastly there is the fake one, both free or with entry fee. The entry fee one often feels like a scam to me. For example, you gather a pool of fees 50K from entrance fee, give winners 10K in award, pay the juries 20K, then benefit from the remaining 20K.

Now as for why we do competitions. For firms, it is almost always after a real project commission. For individuals, the main goal is to sharpen your skill and build the portfolio. The competition is just a topic you like.

Depends on the competition.  Look for comps run by your state AIA chapter.  I've done several of those in the past and while not big projects they where fun, cheap to enter ($25) and if you won you got wee bit of prize money ($1,000 - $2,000).

While I was in school back then (2006-08) we all had this idea that we should get together to participate in some competitions, as a way to get a head start in our careers...

I never did, though, and went on to work in offices mostly developing (invited) competitions.

I wonder if students nowadays still have this idea and are still excited about participating in competitions early on their careers?

Did a couple competitions fresh out of school, fun and all and rewarding, and useful for portfolio, but just couldn’t afford to do any while also working fulltime, my head was too full and my girlfriend wanted to spend some time together as well, so we ended up doing some competitions together, Europan among others, worked out great. Will start an office together after my next job or so.