First, to clarify, you say this only happens on the mirrored instances of SQL Server, not the stand alone instances.

When you get these errors, has there been a failover from the primary database to the mirrored database? If this is the case I suspect that the problem is a difference in the SID for the login between the principle and mirror database. If this is the case you have three choices.

1) create a process that runs on the mirrored systems to fix the SID when there is a failover. This will be needed on both systems.

2) copy the login (master database) from the principal to the secondary. What you really need is the SID. If these match between principal and mirror you don't have to do #1.

3) switch to using a Windows domain account