It depends what you mean by “court costs”. There are fees to file the case, but for the most part the judges time is paid for by taxpayers.

The vast majority of the expense in a court case will be the lawyers on both sides. These are, by default, paid by each party. There are some specific cases where the loser has to pay the winning side’s lawyers, but it’s pretty rare.

There is a token fee to file a lawsuit, but other than that yes courts are funded by taxes. Both sides have to bring their own lawyers.