Can a known recipient of sending via sending stones conceal their identity?

A character has discovered one of a pair of sending stones in a D&D 5e game. They can use the stone to cast sending, with the target being the holder of the other stone. The person holding the other stone, who we’ll call the "recipient", is in fact someone that the character knows, but they are unaware that it is the recipient who holds the other stone, and the recipient does not want them to know. Can the recipient reply to the sending while concealing their identity from the character?

The question of the caster of sending concealing their identity has already been addressed in Is there any way to fake/conceal your identity when casting Sending?, and the caster cannot conceal their identity because sending says:

You send a short message of twenty-five words or less to a creature with which you are familiar. The creature hears the message in its mind, recognizes you as the sender if it knows you, and can answer in a like manner immediately.

This does not address whether the recipient can conceal their identity, but this is because the caster of sending must be familiar with the recipient in order to cast the spell. It’s therefore reasonable to assume that the recipient’s "reply", which is sent "in a like manner", will mean that the caster will recognise the identity of the recipient, because the caster knows the recipient, and because the caster knows who they targeted with the spell.

However, sending stones bypass this familiarity requirement:

While you touch one stone, you can use an action to cast the sending spell from it. The target is the bearer of the other stone.

There is no indication here that the user of a sending stone must be familiar with the recipient, who holds the matching stone. (And such a requirement would make sending stones considerably less useful! So I think it’s reasonable to assume that the user of a stone is not required to know the holder of the other stone.)

However, in the situation I’m describing, the "caster" is familiar with the recipient, but the recipient does not want them to know who the recipient is.

Obviously, the recipient can refuse to respond. However, it’s a lot more fun if the recipient can respond, while concealing their identity. So… is there a way for a known recipient of sending cast via sending stones to conceal their identity?

Use of additional magical items or spells by the recipient (such as the ring of mind shielding suggested in an answer to the linked question) is acceptable in an answer.