Sharing a secret key between many users I would like to know if there is any security issue in sharing a secret key (of an asymmetric cryptography) between several users.

I know that this secret key is no longer "secret", so it's no longer valid for signature.

The advantage of doing that is to allow a user to send a message to many others (all those sharing the secret key) at a time by encrypting it with the corresponding public key.

But I don't know if this will induce security problems.
The only security issues that I can think of when sharing the same key among several users (in an asymmetric encryption system used to broadcast messages) are:


We can't revoke a single user without impacting all the others.
We loose the flexibility to encipher to an arbitrary set of users (which, for large messages and when using hybrid encryption, has moderate overhead).
If the key is determined to have leaked to someone not authorized, we no longer know from which user (or user's device) it has leaked from.
If the ability to decipher messages is determined to be available to someone not authorized, we can no longer selectively disable users to try to determine which user (or user's device) is involved in the leak.
If extracting the key form one device requires making a large number of experiments, and that's practically not possible to an adversary (due to a usage counter, or the time required, or because the experiments are destructive), an adversary in a position to use multiple devices with the same key is in a better position than if each device had a unique key.
This is an issue in side-channel protection of Smart Card (where resistance is quantified in number of power traces allowing to recover the key), and some fault-attacks that similarly require many experiments. As an extreme example, if an adversary could force any chosen key bit to 0 or 1, and when that has changed the key the device self-destructs together with all the key bits, and the adversary can find that, then there is a simple attack finding about $2n$ bits of the key shared by $n$ devices, when there is no threatening attack to any individual device.