I say, Let's ask the recipient of the communiqué: "Did you understand?"

What would an answer of "yes" communicate to the speaker? How would the speaker "know" what the listener understood? How would the speaker "know" that the listener understood what the speaker intended?

Such a question-answer sequence conveys a false sense of security and reassurance; it in no way communicates that there was actual understanding. The only way for a speaker to "know" that the listener understood is by inferring, based on what the speaker hears the listener says in response. We already "know" that exact quotations do not convey understanding; they only convey ability to repeat words exactly, so that's not a criteria. The words the speaker hears the listener say in response must be evaluated and further abstracted and compared to what the speaker remembers he (or she) thought she said, and that evaluation must appear concordant with the speakers desires and intentions (which may or may not entail that the listener "understand"; it may only entail that the listener does what the speaker wants).

"Do you know what I mean?" asks for a "yes" or "no" answer, and this question is no better than "do you understand what I mean?".

Example:
There was an Iowa farm boy who enlisted in the service, and he was serving in a military fuel depot. An order came through which stated, "Rotate the drum stock.". This fellow "knew" exactly what was meant, and he proceded to turn every barrel of fuel oil over. This of course, stirred up the settled sludge at the bottom of the barrels, causing significant maintenance problems for the aircraft using the fuel. Drums of milk in the farm environment needed to be turned over daily - "rotated" to reduce the stratification.

"Rotate the drum stock", in the fuel context meant issue the oldest drum in stock first - first in first out. It did not mean turn them over.

"Do you 'understand' what I mean?" and "Do you 'know' what I mean" both would have gotten the same answer, a resounding "Yes.", but what the listener "knew" did not match what the "speaker" intended.

A question "Do you know what I mean?" cannot be asked "on objective levels". It is a verbal level formulation both when asked and when heard. The farm boy brought a non-verbal level experience (objective level) history from memory that was distinct from what the officer issuing the order intended from his non-verbal experience. No one, except possibly one of conjoined twins sharing part of a brain, has any hope of experiencing another's "object level" experiences or "knowing".

Both questions, "Do you understand what I mean?" and "Do you know what I mean?", do not function to convey information. They only convey reassurance without understanding.

Here's the way to get the desired result.

Don't ask, "Do you understand what I mean?", say instead, "Tell me what you understand I meant.". If the answer is cooperative, then the speaker is in position to evaluate whether or not, or to what degree, the listener understands.

They are communicating perfectly well, but their understanding varies with their experiences. This is an example illustrating that "meaning resides within the person". As Don Kerr said, the speaker's words are heard and understood with the listener's experiential elements. Effective communication depends on pre-established agreement as to the use of words. In the problem case, two distinct universes of discourse used the same phrase differently, and the listener had no awareness that he might be in a distinct universe of discourse.