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These specific Covid-19 vaccines have been tested in humans in clinical trials in accordance with strict FDA guidelines. There were no shortcuts along the way.

As far as long-term research: The research does not stop just because the vaccine in now in use under FDA Emergency guidelines. The initial research volunteers for each of these vaccines are all being followed for the next two years at a minimum. They are still considered "research subjects". If there are no heinous side-effects noted, then they will likely receive full, unqualified FDA approval.

As far as long-term research: The research does not stop just because the vaccine in now in use under FDA Emergency guidelines. The initial research volunteers for each of these vaccines are all being followed for the next two years at a minimum. They are still considered "research subjects". If there are no heinous side-effects noted, then they will likely receive full, unqualified FDA approval.
The vaccine is safe and it works. A brief history of the research and development of mRNA vaccines:
https://www.statnews.com/2020/11/10/the-story-of-mrna-how-a-once-dismissed-idea-became-a-leading-technology-in-the-covid-vaccine-race

https://www.the-scientist.com/news-opinion/the-promise-of-mrna-vaccines-68202