I'm not sure exactly what you are asking.
Is your office machine getting all it's mail from a POP3 server or are you using an Exchange Server for your mail?
What exactly do you mean that a client PC syncs with your other machine?
With Exchange you can delegate authority to another user to access your email account (actually any Outlook function since they all reside on the Exchange Server anyway) and as such, they have either a copy or access to your account to see all the email in that account.
At home that's a bit more difficult to do unless you want to set up an Exchange Server to do the same thing. You can however, have two machines receiving mail from the same POP3 account. They may or may not stay in sync depending on you run the machines.
You can set Outlook to leave a copy of the messages on the server. Depending on your ISP, this will either result in them deleating your oldest messages when you get too many in your inbox, or they will start bouncing messages when your mailbox gets full.
One way around this is to have one of your machines (A) be the one that will download the files from the ISP and delete them while the other (B) simply leaves the original on the server. If you leave B on at all times with a 1 minute time to check for new messages, the B will have most of the messages. The problem is that when you fire up A and download the messages, anything that comes in while A is on-line and B is between it's 1 minute check time will only reside on A. You can always do an import to B from A to keep them in sync.
Is this what you wish to accomplish?
Of course you could set up a small file server and keep the .pst file there. That way you could access it from any other machine you have on the network. Just remember not to open any given .pst file on more than one machine at a time. It gets real ugly real quick.
jdg