I'm thinking about creating an encrypted messaging web application where users create RSA key pairs, they store the public key on the server for others to see and also store an AES encrypted private key on the server. Thus, to verify their identity, they can decrypt the private key and send a signature.
That said, I'm curious is this is a reasonable idea. I know some similar services that require you always use the same device (like an iPhone) and won't let you use the same account on a different machine. This leads me to believe that they are storing a private key locally on the device. It seems reasonable to me to store an AES encrypted RSA private key on the server, but I am no expert on the subject and since I don't see anyone else doing it, I'm a little skeptical.
Any thoughts or concerns about this?