If you cannot even put on your shoes, or they cause you throbbing pain, that is a clear sign that they may be too small for you. If while wearing and climbing in the shoes you cannot actually benefit from them and do things such as toe hooks because of the pain, then that is also a sign that the shoes may just be too small.
On the other hand, if you feel your feet slip and move around inside the shoes while climbing, then that is a sign that they may be too big. Your feet should not really be moving around inside your shoes while climbing, you should feel the support of the tight and snug fit of the shoes. If you do not feel this, they may be too big. If you have a lot of dead space in your shoes, meaning, air pockets between your feet and the shoes, this may also mean that the shoes are too big, or not the right fit for you.
There is a third option, which is that the shoes are just not the right fit for you. Climbing shoes come in all shapes and sizes. Some shoes have a wider toe box, and some have a more narrow heel area. It can be that you will find shoes that fit the size of your heel perfectly, but, the toe box is either too big or too small. This situation is much less about the actual size of the shoe, and more about the shoe not matching your feet.
A climbing shoe normally feels tight, snug, and applies a certain amount of pressure to your feet. This is especially true with new climbing shoes, as they have not yet been broken in. If the shoes are synthetic, they may not have a noticeable break in period, and however tight and snug they are when you buy them, may be how they feel even after a few months in them.
The same climbing shoes in the same size may also feel different for different people, as people have different shaped feet. The same shoes that may be applying a lot of pressure on your big toe, may be causing someone else some pain in another area of their foot. Everyone's feet are shaped differently and have different characteristics, so not all shoes will fit everyone the same, but, you do get used to most of the pressure after a short while.
Once you get used to wearing climbing shoes, a lot of the slight pains, discomforts, and other pressure points start to fade, and your feet get used to them.
You do not normally need to force the break in period of climbing shoes, this will happen naturally by wearing the shoes and climbing in them.
There are of course different methods for breaking in climbing shoes faster, but, they are optional and not required. Some methods could even be harmful for your shoes.
Whether your shoes need to break in or not depends on the type of shoes you bought and the size. If you bought shoes that are synthetic, don't expect any noticeable change even after 6 months of climbing in them, since synthetic shoes don't really have much of a break in period, and if they do, it's very minimal. This should also be a determining factor when buying climbing shoes, since you probably won't want to size down synthetic climbing shoes, expecting them to stretch.
Leather shoes, on the other hand, do stretch quite a bit, so if you bought leather shoes, and you sized down, then you should expect a break in period.
Rock climbing shoes that are too tight can potentially cause damage to your feet, some of the most common being muscle pain and bunions.
Climbing shoes should be tight and snug, but not too tight. Climbing shoes should not cut off or reduce circulation to your feet, and should not cause lasting pain even when taking them off.
Wearing climbing shoes that are too tight can also be dangerous from a usage point of view. If you fall from a route or jump down, landing on your feet, due to the stiffness of your muscles, they may not be able to absorb that impact as well as they should, which could cause damage.
The key to reducing the possibility of damage and pain is to choose climbing shoes that actually fit you and the shape of your feet.
It depends on the type of shoes and their purpose. Synthetic climbing shoes may stretch up to half a size, and even that is not very common. Usually, there is no noticeable break in difference with synthetic shoes, which means that if you go down a size in them, and they are very tight and painful, that may be how they will always be.
Leather shoes, on the other hand, do tend to stretch, and can even stretch up to 2 whole sizes, some even say more. Because of this, many climbers go down even 2 sizes.