Yes, there are several companies operating passenger trains in Poland. However, http://rozklad-pkp.pl/ should be a source of truth and cover all operators - redirecting you to the operator sites as necessary to book. For trips that involve multiple operators, you'll need to buy each leg as a seperate ticket from the respective operator.
For long distance trains, the vast majority are run by the national railway, PKP. Most regional train operators are government run. A few are private companies that some regions have contracted out to.
A couple of private companies are starting non-subsidized "open access" routes, competing with the existing railways. It's not clear to me if any of these are actually operating yet. They seem to be being allowed to start on rather niche routes.