Probably the simplest / safest option is to 'flatten' the PDF before sending it, by printing it to a new PDF.
After completing your form in Preview, go to the Print menu (
cmd+P) and use the PDF drop-down in the bottom-left, selecting to
Save as PDF... to generate a new, 'flattened' PDF.
Flattened simply means that all annotations (including data in the form) become indistinguishable from the actual content of the PDF, as if they were part of it all along. This means you can't edit them anymore (in the new, printed PDF), which can be desirable, but also means they'll show up properly in Adobe.
The actual cause of the problem is detailed here from Adobe, and that blog post also includes a script for Acrobat users to recover the data from a form filled in using Preview.app (it turns out Preview puts the data in the form, but doesn't specify how it should 'appear', so Acrobat shows it as blank/invisible).
I'll leave it up to the reader to decide whether this is a bug in Adobe's software (for not showing form data without a specified "appearance") or a bug in Apple's software (for not including the "appearance" information).
If you've been affected by this bug, please consider notifying one/both of these companies of the issue: