What's the best way to say "Chinese speaker" in the sense of "a person who speaks Chinese"?
I think the challenge lies with how the English word "speaker" does not translate well to Chinese, in the context you describe. 

From the dictionary, we can see that speaker in the sense of language ability simply translates to 說。I would argue that adding 會 or 能，thereby making it 他會說法語 or 他能說法語 , is giving more information than required. Perhaps depending on context, it would be necessary to respond this way, but a simple 他說法語 is a short and complete sentence on its own. 

Also, I would like to point out that there are certain pockets of the world where "Chinese" refers to both Mandarin and Cantonese, and a follow up question to "Do you speak Chinese?", is, "Which one?"  

In certain industries, multilingual employees will wear badges indicating what languages aside from standard assumed (Eg. English in North America, Japanese in Japan, etc.) can be used for conversation. Sometimes, instead of just the name of the language, you can see "精通" as well. While it should translate to proficient, that's often a generous description and should actually be taken to mean "conversational in". Truthfully, "通曉" would be a better description. 

On that thought, "通" could be what you're looking for if you're looking to describe someone's ability on paper. Think of an online dating site. A possible description could be: 女，25-35歲，通國、粵、英語。

Going back to the problem of "speaker" not translating well to Chinese for a moment, notice that there isn't a single character to mean "person orating" or "person giving a speech". See the dictionary link above. In Chinese, you typically need three characters; 演說家、演講者、說話者, with the last one being awkward but proper in my opinion.