Is there a way to make issues publicly viewable in JIRA for a particular project? By public I mean that it is not required to login to view them. All other projects should stay private.

Create a specific permission scheme for your 'public project' Grant anyone browse rights <img src="https://i.stack.imgur.com/lhbwR.png" alt="Choose group 'Anyone'"> Associate the new or modified permission scheme to the project(s) you want to open up. That's it. Francis