The mRNA is translated to a protein, which is the actual target your immune system then recognises as foreign, attacks, and remembers.
In a sense, it's a lot like a prodrug, which is a "traditional" small-molecule that is turned into the actual drug by some interaction with your body–your stomach acid, for example.
"mRNA" might sound scary. But it isn't going to change your genome or anything like that; it's strictly downstream from DNA. Any virus, bacteria, or salad you eat contains far more mRNA in addition to DNA or RNA, and the first of those really does come with mechanisms to insert itself into your cells' DNA and has evolved to defend itself against your defences, and to slowly take over your body in some way or other.
There are some short-term risks on the order of, say, eating lobster for the first time. Allergies, for example, which can be quite severe and even deadly. But those are exactly the risks the trials would have uncovered, and the people possibly at risks of such things would already know they are susceptible.
There is no known mechanism for any risk that would not manifest within the first day or two.