Conclusion first, neither term is particularly common.
Ironically, however, places like this (SE) are not very good to discuss the matter. Why not? Because the Japanese-speakers who are on here, myself included, are already at least bilingual to an extent and, in general, they would tend to be more interested in foreign languages or "Language" in general so that they would spend more time discussing language-related subjects than the average Japanese person. For that reason, someone might jump on me saying that my first sentence is wrong and we use those terms all the time.
The first time I had ever heard the word 「ネイティブスピーカー」(it is one word in Japanese) was when I was in high school. That was around the time when some large-scale English conversation schools had emerged and started advertizing on TV. 「ネイティブスピーカー」 was their catchword; they hired those people. Good-bye to the old days of learning English from Japanese teachers.
Funny thing, in restrospect, was that the word was practically never used to refer to a native speaker of any other language -- particularly one of Japanese. Just as in child language acquisition, over-generalization and over-simplification do often occur when a culture imports words from another.
「[母語話者]{ぼごわしゃ}」 is rarely used outside of those who are interested in linguistics or a particular foreign language, either. I do not think I myself had ever used it actively before I was in college. It is almost like a technical term for linguists.
One could say one would hear 「ネイティブスピーカー」 more often than 「母語話者」, but that is largely because of the advertisement by Eikaiwa schools. The truth, however, is that neither one is used in the Japanese-speaking world as often and widely as "native speaker" is used in the English-speaking world. Japan is not an immigration-based country like many English-speaking countries are. We as a nation just do not have the custom of asking each other on the street the question "What's your native language, man?"
Another (perhaps more linguistically convincing) reason that we do not use the two words in question is that we use the word 「[母国語]{ぼこくご} = one's native language」 way more often. If you use it, you do not need to use 「ネイティブスピーカー」 or 「母語話者」 because you can say 「X語を母国語とする人」 instead of saying 「X語の母語話者」 or 「X語のネイティブスピーカー」. It is only a matter of what sounds more natural in either language.