setting up a local network

I want to have a mini-network that includes my old eMac and my new iMac. I have listed both users (myself with different versions of my name) and in 'sharing' have included most options, allowing both users to read and write. Each tells me that I can access the particular computer under 'name.local'. But I don't see anyplace to put that. Neither am I really sure what I should expect - should I be able to go to the 'name.local' for the eMac when I am on the iMac, and see whatever is on that computer? Any help with this would be appreciated.

Raclare

Using the "Connect to Server" option from the "Go" menu in the finder would be the place to enter the name.local.

You will be prompted for a username and password of a user from the computer on name.local

Thank you. What happens is that I get a folder that tells me I cannot open it because I am not authorized to do so. In the 'sharing' screen I have given all the permissions except those that were not allowed - or advised against. Using the eMac, I can access the iMac, although there is very little there. Last night, when I went to shut down the eMac, I had a warning that the sharing user might lose unsaved data. I went to it - it is like a huge cube, rotating to a new face, and found the flash drive icon. The flash drive was connected to the eMac, not the iMac, but I dragged it to the eject symbol, went back to the eMac, and it let me shut down. I seem to have a one-way network, but want it to work both ways.