No, or else we wouldn't understand the monsters when the episode would cut to Ludo's castle. Okay, obviously that's for audience convenience, but in the Mewni marketplace we see that the signs have both English and Mewnian writing. I also find it hard to believe that Star speaks another language, both because she seems like she'd have shown it off before and because I can't imagine a Mewman royalty that sees itself as superior would be speaking in another place's language the majority of the time, if they already had a spoken language of their own.
My dumbass headcanon is that Mewmans speak English, whether because they were originally from Earth or they borrowed it from there centuries ago or what, and so use the Mewnian writing to represent the same sounds of English speech in a unique way, so that they have some claim on having made the language their own. Both scripts mean the same thing, but being able to read the Mewnian one identifies you as a Mewman.
Maybe think of it like English is their hiragana and Mewnian their katakana. Not an exact analogy since Mewmans seem to use both interchangeably, but pretty much how I imagine the difference between the writing systems.