Here are three different solutions I found:
1) Download Adobe Reader in the language you plan to fill in your form.
Stupidly, Adobe Reader by default seems to come only with a single spelling dictionary (in my case German). So, a second Reader (with the correct dictionary) should help.
(I did not test this myself because I do not have the right to install software on my office computer)
2) Install the Adobe Reader Spelling Dictionary Pack
For Adobe Reader XI:
Adobe - Adobe Reader : For Windows : Adobe Reader XI Spelling Dictionary Pack
For Adobe Reader DC:
Adobe - Adobe Reader : For Windows : Adobe Reader DC Spelling Dictionary
In my Adobe Readers (both XI and DC), this did the job. Puh!
3) Change the settings, then shut down Adobe Reader and restart
This is silly, but this seems to work. I observed this behaviour in my Adobe Reader XI.
Change settings in Edit --> Preferences --> Spelling, then shut down the programme, and restart; then the changed settings seem to work.
However, unfortunately I discovered this just after I installed the language pack mentioned above, so - no guarrantee that it works in a "naked" Reader, too.
Later additional experience: This method worked for me also in Adobe Acrobat X Pro: I changed the language for spellchecking (Edit --> Preferences --> General --> Spelling). This had no effect at first. I then closed the programme, and restarted it, and now the newly selected language was applied to spellchecking the contents of the pdf form.
It took one and a half hours to find these solutions. What a mess. Couldn't they just add a button "download more dictionaries" to their settings? And of course make them work without restart?