HFT shops compete for real estate and hiring (especially this and fiercely - non-complete clauses and lawsuits are standard) with each other but in terms of their trading competition it varies widely based on what algos they use.

An algo is basically just a trading strategy (sure, there's a basic set of these) but speed of data collection/processing and execution gives you more options. You can design an algo to hunt other HFTs or not. You can also trade against HFTs if you're not.