First, welcome to the RStudio Community!
To answer your question, nothing, but it also depends on the application. Are you using any web analytics or trackers? Are you storing user information internally or sending user activity to a third party? Are you using local storage on user’s device or are you sending the data to a self-maintained database? You can also find some of this information by opening the developers tab in your browser and navigating to the “Storage” tab.
I'm assuming you are referring to GDPR?
If your site is part of a larger organisation, check the main organisation website to see how it is implemented there and follow their design.
Since we also talking about compliance, I would also like to point out that web accessibility is an important aspect of creating an application. There are EU guidelines to meet web accessibility standards (possibly organisational requirements too). There are different levels of web accessibility and it would be best to follow internal guidelines. See the W3G site for more information on the guidelines. I would also recommend checkout out the a11y project and W3G tutorials for implementation and checklists.
Hopefully this provides a good start. There’s a lot of information and some of it may not be relevant. If you have any questions, I’d be happy to help.