"Do market makers in the stock market run into this problem?" Yes, that's why they have to account for the probability that whoever hits their bid is an inform

"Do market makers in the stock market run into this problem?" Yes, that's why they have to account for the probability that whoever hits their bid is an informed trader (vs. uninformed/noise trader). And it's part of the reason they can provide price improvement when they source order flow from retail brokers, whose customers are mainly uninformed/noise traders.