Rule 11 sanctions are normally triggered when the offended party files a motion. The alleged wrongdoer gets 21 days to fix the problem, else the court gets involved. Courts can also prompt sanctions on their own—or “sua sponte.”
If it’s a party that brings up sanctions, courts can order monetary fines to be paid to the court under Rule 11 (not to the other party, for fear of creating incentives to seek sanctions as a moneymaking enterprise). Courts cannot impose money penalties on parties (e.g., the Trump campaign) for failing to understand the law. But parties can be ordered to pay monetary penalties for proffering baseless facts.
Moreover, courts can sanction parties and lawyers in ways that do not involve money. At the federal level, non-monetary sanctions have included public reprimands, orders to undergo legal education, referrals to the bar for disciplinary proceedings, warnings, suspension or even disbarment, forced admission of the facts alleged by the other side, and bans on a party or attorney (e.g., Powell or Giuliani) from bringing similar suits without advance permission from the court.