Evidently, tones are a matter of concern for many learners of Chinese (see for example, this question on remembering tones). While learners struggle to describe the properties of tones (is the 3rd tone really falling and rising?), natives as well as many that choose to immerse themselves in sound more so than writing and theory can effectively bypass thinking about tones altogether and go about their days speaking Chinese.
Native speakers: If put to the test, could you write out pinyin with tones?
Non-native speakers: As your Chinese learning has progressed, do you still intentionally analyze the tones of words you learn by speaking (e.g. remember 拼音 has two first tones)? Do you even think about things like tone sandhi (which you may have learned complicated rules for at some point in your studyfing)?
My impression is that (native) Chinese speakers have a grasp of rhyme (initials + finals, as is represented in bopomofo and pinyin spelling to a degree), but that explicit tonal knowledge is limited. This is reinforced through the current way most people write Chinese on the Internet, i.e. through pinyin input which does not require tones. Learners, on the other hand, are likely to be able to easily name the tones for many words they studied via pinyin with tones.
Does this impression capture some generalizations about knowledge of tones among different types of Chinese speakers?