What was the appropriate response if an Early Romantic era composer dedicated a composition to you?
Well of course it depends on who you are, what your existing relationship is, and on what you think of the composer.
Wikipedia helpfully has a category page for music with dedications! Let's take a look:
Africa by Camille Saint-Saëns was dedicated to Marie-Aimée Roger-Miclos. The piece was written in 1891; Saint-Saëns was born in 1835 and Roger-Miclos in 1860. This appears to be a case of a well established composer dedicating a piece to a well established performer. Saint-Saëns had promised Roger-Miclos a new composition, and Roger-Miclos played its première performance. Surely this is the appropriate response in this circumstance.
Ballade No. 4 by Frédéric Chopin, dedicated to Baroness Rothschild, who "who had invited Chopin to play in her Parisian residence, where she introduced him to the aristocracy and nobility." I infer that Chopin composed played the piece in preparation for this performance, in which case the appropriate response was surely to applaud. In addition, since this is a dedication in thanks for patronage, the dedicatee might also appropriately pay the composer an honorarium or take other steps to support the composer's career.
Beethoven dedicated his second piano sonata to Joseph Haydn, one of his teachers and perhaps the most senior figure in Viennese musical life. I did not see any indication that Haydn ever performed the piece but he certainly may have, and it's hard to imagine that he never played it at all. In any event, the undeniably appropriate response here is to accept the dedication as an expression of gratitude.
In one of the subcategories you will find Robert Schumann's Myrthen, dedicated to Clara Wieck, his fiancée, on the occasion of their marriage. The title of the first song is even "dedication"! (More precisely, it is the German word for it, Widmung.) In this case, the appropriate reaction is, um, marital.
Schubert dedicated his ninth symphony to the Gesellschaft der Musikfreunde, which "made him a small payment, arranged for the copying of the orchestral parts, and at some point in the latter half of 1827 gave the work an unofficial perfunctory play-through."
I had trouble finding unwanted romantic dedications, likely because composers tended to rescind them as Beethoven rescinded the initial dedication of his third symphony to Napoleon. An appropriate reaction to an unwanted dedication would be to inform the composer of one's lack of romantic interest.
To reiterate: as with any social interaction, the appropriate response depends on the circumstances.