Should you be able to wiggle toes in shoes?

You should not be able to free wiggle your toes within climbing shoes. Your toes should be slightly bent downwards to give you better accuracy and the ability to apply the required amount of pressure and force from your toes when standing on small foot holds. If your toes wiggle within your shoes, you may not be able to actually apply the amount of force you need.

If your toes freely wiggle within your shoes, they may be slightly too loose, or the shoes may just not fit the shape of your foot.

Generally, no. Climbing shoes are supposed to be pretty tight.

I wear climbing shoes slightly bigger than my street shoe size, and I still cannot wiggle my toes.

I can slightly move them up and down a little, but not so much that I would consider it wiggling them.