Should climbing shoes be bigger or smaller?
Climbing shoes should be snug and tight, but not too small. What that means is that there should be as little dead space in the shoes as possible, ideally, none.
Climbing shoes are designed to be snug and tight, and to not have dead space within them. But, climbing shoes come in all shapes and sizes, and not all shoes will fit every climber. This is why it is important to try on different shoes and different sizes in order to find the right pair of shoes for the shape of your feet. The right pair meaning shoes that are snug, have no dead space within them, and fit the shape of your feet also in terms of the toe box and how wide it is.
Whether climbing shoes should be bigger or smaller depends on what you are comparing them to. I personally wear half a size larger than my street shoe size in all my climbing shoes. I have a very difficult time finding shoes that fit properly, especially in the heel area, and I find that going up half a size is even a semi performance fit, as going down is just unbearable.
So you can go down in sizes or even up, as long as the shoe fits the shape of your feet, is snug, tight, and has little to no dead space, and most importantly, is not too small.
If it helps, I wrote an article on this very subject on this site's blog, it goes over all the steps for choosing the right climbing shoes for your feet: Climbing Shoes 101: How To Pick The Right Pair For You