In some instances of private/public key systems, this is done. It’s mainly for the purpose of ensuring the recipient knows who the sender was and also ensuring the sender knows who the recipient is.
Quick primer: If you encrypt with your private key, everyone knows it was sent by you. If someone encrypts with your public key, they know you will receive it. Use your private key and someone’s public key together and you know only that person got it.
In practice, lately another step is added to negotiate a third temporary/session key. This ensures keys aren’t used forever, and if compromised a new one can be generated. This is more secure than encrypting twice, because you never know what data is sensitive and picking the wrong one requires the attacker to start from scratch.
Pagers are not guaranteed to be 1 way comms and bringing them into secure locations is a security violation. Additionally, depending on the classification, no unauthorized and undisclosed devices of any kind would be permitted, including any electronics or electronic media such as tapes, CDs, discs, etc. Even when I was issued a verified 1-way pager, I was specifically briefed I was not permitted to bring it into a classified location. Most of the highly classified SCIFS are shielded anyways, you can’t use it inside so it’s safer to leave it out, along with all other devices.
If your organization allows it, then (if federal) they are breaking the law and should be reported/up-channeled. If it’s corpo, you should bring up additional concerns with your security team.
Edit: Also, it goes without saying, current events are probably a good reason why pagers (and other devices) aren’t allowed in classified areas. While most focus on disclosure (getting out), we must not forget the risk of data/operations getting destroyed.