Adobe Acrobat (Standard and Pro) can change PDF forms to enable Adobe Acrobat Reader users to 'fill+save' form data (instead of the standard 'fill+print').
It is a special option available when saving the PDF saying "Save PDF with extended Reader functions" (or similar... I'm translating this back from German into English).
This cannot be achieved with any non-Adobe PDF creating software (unless this has licensed that function from Adobe). It certainly cannot be achieved with LibreOffice.
The technical reason for this is that Adobe uses a digital signature to protect this function, and that you'll have to agree to not reverse engineer the key when you accept the Adobe EULA.
Acrobat Reader has that key compiled into its binary. If it checked and verified the key, it will change the message displayed to the user indicating that the form data of this document can be saved (it will also change its behavior and indeed save the data).
On the other hand, it looks like most non-Adobe PDF readers will be able to save the formdata, and read it correctly from the saved file. (Adobe Reader however will refuse to read the data, because the saved PDFs lack the required signature!)
Not sure if these infos will help you, but they reflect the real life... :-(