Please help us solve our debate. One person says to write "Jess' Bridal Shower", the other says to write "Jess's Bridal Shower". What is the correct way, or are the both correct?
This is a question of style, and different style manuals will give different recommendations. For example, both the Chicago Manual of Style and Garner's Modern American Usage would recommend Jess's Bridal Shower. On the other hand, the Associated Press Stylebook would recommend Jess' Bridal Shower.
So, arguably, if this is really a point of contention, then what you and your friends should do is this: first, agree on the style manual to use in all text connected to the event. Then, when there are points of disagreement among your friends as far as writing, you will all agree in advance to follow the recommendation of the agreed-upon style manual. In other words, the manual will be your 'arbitration official' in cases of disagreement.
I understand that you may not have access to any of these manuals. However, you can always post a question here on ELU, of the form According to [the name of your chosen style manual], how should one write...
For more details on the general rules about forming possessives given by all these manuals, see this answer.