We recently set up Outlook Express, with Multiple Identities. 
However, the identities continuously receive the wrong mail.  Each
accunt is set up with a different e mail address and user name.  In
addition, "include this address when receiving mail or synchronizing"
is checked, for each identity.  "Remember password" is also checked;
we do not want to use passwords every time we sign on.  WE've tried
checking and unchecking "Send and receive messages at start up",
however neither seems to make a difference. To switch accounts, we
always select "Exit and log off identity".  Our server suggested
deleting all the accounts and re-uinstaling them, but I don't want to
go through all that again. What should we change to correct this
problem?

I had the same problem - the answer is to go to Tools then Message Rules.

Switch to the identity for person1@ISP.com and create a Message Rule stating
"Where To or CC contains person2@ISP.com, Do not download it from the server".
Place this rule at the very top of the list.

Then do the same thing for person2, to exclude mail intended for person1.

Hope that helps - it worked for me.