Yep, the municipality decides on such matters. Places do still occasionally have their names changed (rarely of course, because it involves a lot of work including updating addresses), usually aligning with local use. In the case of De Westereen a name from the local dialect replaced both the Dutch and Frisian names (Zwaagwesteinde and Westerein, respectively).

In a number of cases originally Frisian names actually supplanted older Dutch names (e.g., Burgum, Grou, Eastermar, etc.), so those places have just one name in both languages (except on the Dutch language Wikipedia because of weird reasoning about allowable sources and apparently a hatred of Frisianised Dutch names).