Windows file sharing is obviously the best solution for long term file sharing between two machines - however, it can be problematic on a temporary / one off basis.
I would highly recommend that you download Filezilla Server and install on the old machine, then set up a new user with a username/password of whatever you want, and share the root of the hard drive as the home folder.
On the new machine, simply type ftp://xxx.xxx.xxx.xxx (where the x's are the old machine's IP address), and then authenticate and copy the files over.
If however you do want to copy files still using Windows sharing, you probably have set the share to read only - you can try right clicking and going to
properties then looking on the security tab and temporarily enable
everyone (Or just the user if you know who/what it is) to have full access. You will also need to do this under the sharing tab.
However, I have to say, it is always easier to copy from a share than to a share, e.g. use your new machine to browse a share created by your old one - this is mainly just a personal preference.