I do not handle these sorts of cases, but I am familiar with these fee arrangements. This kind of legal arrangement is called is called a contingency fee.
Typically, the person suing will sign an agreement with the attorney handling the case under which the person pays the attorney's fees for representation directly from whatever the proceeds ultimately will be from the case. Usually the person suing has to front any expenses, however. Washington State ethical rules do allow the law firm or attorney to loan the client money to cover these expenses, but this is pretty rare.
The percentiles payable in final attorney fees can vary a great deal depending on the type of case, the difficulty of the subject matter, and when the matter is ultimately resolved. I have seen contingency fees similar to the following: 25% paid to the lawyer if the matter settles before filing a lawsuit, 30% is it settles after filing suit or during a trial, 40% if there is no settlement but it goes to trial and you win, and then if it gets appealed and the verdict us up held, 45%.
The concept is that unless you win, you will pay nothing in attorney fees.