The normal Globus mode of operations is to use endpoints at either end of a data transfer. An endpoint is usually a dedicated data storage system with high performance network interfaces and disk storage, tuned for long distance data transfers.
It is also possible to set up a personal computer as a Globus endpoint by installing Globus Connect Personal software on it. Note: that this will probably not perform as well as a dedicated, tuned endpoint. If you have a local server-class Globus endpoint at your institution, it will likely be better to transfer data to that endpoint, then make a local copy down to your own machine.