Well, some jewelers just mangle things. Most don't, though, and any jewelry store that isn't full of ham-handed idiots can do that with no damage to the finish and they'll charge you little or nothing.
If you want to do it yourself, push the pin out in the direction indicated by the arrow, and replace it pushing it in the same direction -- do not reverse that, always push the pin in the direction indicated by the arrow. So you push it in the side it didn't come out.
It's sometimes necessary to use quite a bit of force with a brand new bracelet. You might have to tap on the tool you use with a hammer or something -- don't pound with great force, just tap on it and it should come. If you have to use a lot of force it's important to use a pin punch the right diameter -- if it's too small it'll expand the end when you pound on it; if it's a little too big it'll enlarge the hole. Normally it'll come without that much force and anything handy will work as a pin punch -- a paper clip or whatever.
If you have it done at a jewelry store that'll loosen up the pins so you'll be able to make future adjustments yourself.
If you're into gunsmithing at all you'll need pin punches. You can buy a set but most of us make our own with broken drills or any rod we happen to have around -- if you don't have the size you need you can buy a drill and mount it in a dowel for a handle. Avoid high-speed steel drills if you're buying it for that purpose -- get a cheap carbon steel drill and heat it with a propane torch or over the stove to soften it. A pin punch doesn't have to be particularly hard and drills are so hard they're brittle and break easily. Brass rod works if you happen to have some the size you need.