What is a good hardware setup for using Python across multiple users
Is installing Python locally a good practice?  Yes, if you are going to develop in Python, it is always a good idea to have a local environment where you can break things safely.

Is there value in setting up a Python "server"?  Yes, but before doing so, be sure to be able to share your code with your colleagues using a version control system.  My reasoning would be that, before you move things to a server, you can move a great deal forward by being able to test several different versions in the local environment mentioned above.  Examples of VCS are git, svn, and for the deep nerds, darcs.

Furthermore, a "Python server" where you can deploy your software once it is integrated into a releasable version is something usually called "staging server".  There is a whole philosophy in software engineering &mdash; Continuous Integration &mdash; that advocates staging whatever you have in VCS daily or even on each change.  In the end, this means that some automated program, running on the staging server, checks out your code, sees that it compiles, runs all defined tests and maybe outputs a package with a version number.  Examples of such programs are Jenkins, Buildbot (this one is Python-specific), and Travis (for cloud-hosted projects).

What are the hardware requirements for such a box? None, as far as I can tell.  Whenever it runs out of disk space, you will have to clean up.  Having more CPU speed and memory will make concurrent builds easier, but there is no real minimum.

Do I need to be concerned about any specific packages or conflicts between projects?  Yes, this has been identified as a problem, not only in Python, but in many other systems (see Dependency hell).  The established practice is to keep projects isolated from each other as far as their dependencies are concerned.  This means, avoid installing dependencies on the system Python interpreter, even locally; always define a virtual environment and install dependencies there.  Many of the aforementioned CI servers will do that for you anyway.