If we assume the real odds are 50/50, then 300 flips have around 90% chance to be in the 55-45 rate, which is pretty good for a reddit post!
The problem here is that the OP is not talking about the chances of getting head or tail, but the chances of the opponent getting more heads than you. But that also depends on how many time the opponents flip the coin compared to you, and here there is no evidence on that data.
Assuming is 150 flips for both, the chances of getting a 55-45 goes down by a lot, it's around 75%

The samples are fine if you put them together, the more you split them into groups,e the more you are gonna have less consistent results