Climbing shoes should be snug enough to not have dead space in them, but not so snug that you feel throbbing pain and can't even walk or climb in them.
Climbing shoes are designed for being snug and minimal to help with footwork, this includes standing on tiny holds, and pulling your weight with your feet. If your climbing shoes are not snug and tight enough, your feet may move inside of them, which can make it much more difficult to pull your weight and stand on small holds.
Listen to your body and your feet, if it's too painful, go up a size or find different shoes that fit your feet. Having extremely tight shoes is not the answer if it reduces your performance.
