For the first puzzles (all blue pieces) it doesn't really matter. You can use Nintencoins to buy pieces and help complete the puzzle but it's completely random, so there's no "correct" way to start building the puzzle. Just pick any piece you want.
For the later puzzles, however, purple pieces take priority over everything. If you run into someone who has a purple piece of whatever puzzle, take that one. Who knows if anyone else has that piece or if you will ever run into that person again.
I also try not to use coins if I have puzzles that are almost complete and know someone who has these pieces.