Again, catastrophic floods happen all the time. The theory suggests that the Black Sea flood happened in 5600 BCE. The earliest version of the Epic of Gilgamesh we have is from 2000 BCE and the king it was likely based upon lived in 2700 BCE.

Now what is more likely, that a big flood happened not all that close to Sumeria and people kept talking about it for thousands of years or they just lived on the Tigris, which floods all the time and that's where the story came from?

If nothing else, Sumeria was nowhere near the Black Sea. It doesn't even make sense geographically.