Climbing shoes should fit snug onto your foot with your toes curled. It will feel tight so your feet and toes cannot move around. It is important to not have internal dead space or gaps between the toes, as they can reduce the sensibility. Usually, when the fit is tighter, the performance of the product is higher.
The shoe material can change the perception of the foot and the performance on climbing, as the types of climbing you want to approach.