Hi María,
All participants in the same study batch share the same bath session data, so giving people a different link will not affect this.
If your study is longitudinal and you need to store data from one session to the next, then the batch session data is what you need
If you just want to share data between components, I would use the study session data, which really should work (because, to answer your specific question: yes, you might also have concurrency issues). Using the study session data in this case is the clean and best solution. If I were you I try to set the study session data. If that doesn't work, have a look at any possible error messages displayed on the Developer Tool's console. If you can't see anything obvious there, you can always send us a minimal example of your study where you can reproduce the problem and we'll figure it out.