mRNA will not give you the virus.
while getting vaccinated if someone giving the vaccine may have the virus and be capable of infecting someone, but from what i've seen most clinics are being run safely. if you show up to get vaccinated and the site is not being run safely, ask to see the director/manager and voice your concerns immediately, or leave and say why you are leaving.
otherwise, no, the mRNA vaccines will not infect you with the virus.
as to issues of the vaccine masking further infections from being noticed or detected.
i have not yet any huge studies (more than 30,000 people enrolled) of reinfection rates, but that one i have seen indicates that there is a very low risk of reinfection (which has been noted already during the pandemic since it started there have not been a huge number of people who have been getting reinfected). no matter if you have the vaccine or not this risk of reinfection exists, but it is not as of yet a major trend.
you can have this same issue of being asymptomatic from any regular infection too along with that then allowing you to be reinfected later and also having that being asymptomatic but still spreading the infection.
here is a video that is recent, and the guy seems mostly reasonable in his analysis here:
Dr John Campbell - Reinfection rate, very low
and no, i don't think he is perfect in all details in some of his other videos and interviews i've watched, but he has a huge library of videos if you are interested in medicine in general too.