How do I earn most coins in Swarm when competing with others
First, be diligent about checking in.
Second, use the stickers (and if you have enough coins, make sure to upgrade them to the 2x and 3x bonuses), but use them strategically. For example, if I know I'll check into a place or category multiple times that week, I will try to maximize my bonus from the sticker use. If you're attending an event (e.g. concert, movie, etc.), make sure to select the event (if it allows you) after checking into the location.
There are certain situations in which you can earn bonus coins, including first time checking into a place, first of your friends to check into a place, new states/countries, X miles away from your last check in, X times at a specific place or category, being a mayor (or even better becoming a mayor), etc. I try to align my sticker usage to maximize these situations.
If you know you are close to becoming mayor of a place (can find this at the bottom of the mayor screen), try to use the sticker for when you dethrone. Oddly enough, sometimes it actually helps me when someone dethrones me of a place that I frequent because I end up becoming mayor again within a few days to a week (and earn the dethroning bonus).
PS. Get bonus coins for photos attached to your checkin.