This issue comes up fairly frequently on here, try searching the threads and you will find plenty of discussion on the pros/cons of fellowship.
I chose to not do fellowship, I have been in private practice for one year now. That is my bias, so most people like me will say that there is no need to spend another year being told what to do, being treated like scum, and missing out on a significant amount of money in order to do a fellowship.
On the other hand, people who did fellowship will tell you that it was the best decision they could have made and that it set up their awesome career, doing cases they love.
The good news is that in anesthesia, you can find a great career either way, but it depends on what suits you. The hard part is that we tend to train in big academic centers where many attendings did fellowships and where there are strong incentives to influence the residents to sign up for fellowship. It's an echo chamber. Looking back, most of my academic attendings did not have an accurate sense of what it is really like to be in private practice. The ones that did were the few who had actually spent some time in private practice before returning to academia. There weren't very many of them.
Not once have I regretted my decision. Some of my partners are fellowship trained and we have subgroups that cover peds under age 2 and open heart cases. Everyone does everything else. Other than figuring out which partner to assign to do those special cases, no one cares who did fellowship and who didn't.
About half of my residency class ended up going into fellowships. I was recruited for every fellowship at some point by someone in residency, which I think was fairly common at my institution unless you were universally considered to be completely terrible. Politely declining interest was good enough for some people, but others tried to push a little harder or bring it up frequently. But I just knew fellowship wasn't for me.
Personally, I had plenty of job interviews and job offers, no one challenged my decision since they were obviously considering me as a candidate for a general position, not to fill a specialty need. In that regard, if the group you hope to join needs a specialist, then you will be glad you did fellowship. Conversely, if you did a fellowship but the group doesn't need your fellowship level skills, they might pass you by. If you are a cardiac trained, most people assume that you actually want to do cardiac cases and if I can't offer them to you, then I also assume that you are going to jump ship once you find another job where you can do hearts. In this scenario, fellowship can be a significant liability. But then again, if that perfect group really needs to cover some cardiac cases, then you are golden. This is why the best strategy is to pursue fellowship if you are either going into academics for sure or if you truly love that part of anesthesiology because it is difficult to project if it will help or hurt your prospects at specific private practice groups since hiring needs vary from year to year.