Those required sample sizes serve different purposes.
In frequentist approach you start with presumable effect size, then choose statistical significance and statistical power levels, then compute sample size. An interpretation is following "for effect size equal or greater to your chosen value, you would pick 'best' group with probability related to your chosen statistical significance and power levels". (I might be wrong here, but I think it's something like that).
So, if you want these guarantees on your probability to pick best group, you should run your experiment at least as long as your calculated sample size.
Bayesian approach typically operates differently. It works with data from a single experiment. Given your current data, it estimates probability distributions of your metrics and related values. For example, probability that metric in one group is "greater" than in other P(conversion_b > conversion_a | data ). There could be different conditions to stop an experiment, but let's assume that experiment is stopped when this probability reaches certain level, say P(conv_b > conv_a | data ) >= 0.95. If your current probability is lower P(conv_b > conv_a | data ) = 0.7 then you can simulate how much additional data you need to reach the required level. If you do not have data at all, you can run these simulations using prior distributions. So, this is not a required sample size, but rather an estimate of how much data you need to reach your stopping criteria.