Handel, Bach and Couperin were not at all happy with ornament improvisation in their pieces, and went to great pains to give very specific directions on how they wanted their pieces ornamented. Performers of the time were livid, since this was their prerrogative ("Composers" hmpph! Who they think they are!
). On the other hand, it is thanks to these three that we today know much about Baroque ornamentation practice.
Scarlatti on the other hand, either could not be bothered with providing unambiguous directions, or he truly did not care (and was perhaps curious and excited to see what perfomers would come up with). He uses ornament signs like "tr" and the wavy line, as well as little notes, but nowhere does he tells us if the little notes are apogiaturas (and how long they should last) or acciacaturas. tr and the wavy line are used inconsistently, and we never know if the ornaments start on the main note or not, nor for how long. In short it is all informed guess work.
Scarlatti was a prodigious virtuoso of the keyboard (there are accounts of a famous duesl between him and Handel, where Handel was declared the best of the organ, but Scarlatti was considered unsurpassable on the harpsichord), who innovated deeply and widely on keyboard technique. So it is very likely that he himself improvised not only the ornaments, but most certainly in the sonatas themselves, and it is well possible (the same case has been argued for Mozart sonatas) that the scores that came down to us are just "skeletons" or guidelines, in the same way that fake books just give the outline for the jazz musician to improvise.
So, yes, I would say that one should improvise ornamentation if one feels prepared and inclined to do so.
Best wishes,
Bernhard.
Topic: Scarlatti sonata in D minor K 52