The term chromatic mediant refers to a relationship that can apply to chords and to keys. If the roots of two chords are a third apart and the qualities of those chords are the same, then the chords are chromatic mediants of each other; if the tonics of two keys are a third apart and the mode of those keys are the same, then the keys are chromatic mediants of each other. Your question is impossible to answer directly beyond just saying, “it depends entirely on everything else that’s happening.”
Let’s say I have a piece built on a chord progression like this: C–E–Amin–Dmin/F–G7–C. It depends on other aspects of the composition, but this progression could work well as the basis of a common-practice tonal piece in the key of C Major. Yes, the second chord is a chromatic mediant of the first chord, but that doesn’t make it non-functional—in fact, it’s just the applied dominant of the Amin chord. The whole progression is the entirely functional (and only slightly chromatic) I–V/vi–vi–ii6–V7–I. There’s no key change just because a single chord was chromatic; it would almost certainly be ludicrous to call it that.
Now say I have a piece with the following chord progression: C–E–C–Dmin/F–G7–C. Again it depends on context, but this is pretty likely to be the basis of something in the key of C Major. The E chord is a chromatic mediant away from the the tonic chord, and now we could potentially call it non-functional (or differently functional). It’s fairly likely that the E chord has more of a purely voice-leading function, and this is something that chromatic mediants are nicely suited for since they have a common tone and the other two chord tones don’t have to move very far to get to new chord tones. Precisely how the progression would be analyzed is more debatable and ambiguous than my previous example, but it might be something like I–III#–I–ii6–V7–I. Depending on context and analytical purpose, a lot of people wouldn’t even apply a Roman numeral to the second chord at all, while others would have different approaches entirely, but if the E chord is truly just a single chord, it’s hard to imagine any serious argument that it’s a key change. Just like in the previous example, the chromatic mediant is just a single chord in the middle of an extremely straightforward C Major progression.
Now let’s pretend we have a piece that makes constant use of this chord succession: C–E–C–A–C. This is completely outside the norms of common-practice, and, depending on context, it probably wouldn’t be said to be in a major or minor key at all. C is probably a tonal center of some sort, just based on its repetition and its prominence at the beginning and end of the chord sequence, but the lack of anything like a proper V chord combined with the fact that neither of the non-C chords belong to a C Major key destroy the reasonableness of a C major identification. We would probably point to the chromatic mediant relationship as the main explanation for the sound of the succession—and part of why it “works”—but we probably wouldn’t say each chord is a different key. What would that even mean? As already stated, the series of chords isn’t in C Major in the traditional sense, but it’s even less in E Major or A Major. A piece built on this chord succession just isn’t in a key at all in the traditional sense.
Of course, it is possible to modulate to the key of a chromatic mediant. The P section of Beethoven’s “Waldstein” piano sonata is in the key of C Major, and the S section (of the exposition) is in E Major. However, it isn’t a modulation because it’s a chromatic mediant, it’s a modulation because the music of the later section behaves like music in the key of E Major. B chords behave like V chords, F#min chords behave like ii, etc. In the “Waldstein,” the chromatic mediant relationship of the keys is definitely noteworthy, because it was a very unusual key relationship at the time, but it isn’t why it’s a modulation.
Just using a chord isn’t somehow a change of key, a key change requires a shift in the grammar of the music.