What you see in Object Explorer, is actually a connection to the server using your local machine credentials, not any SA credentials. When you click on the server in registered servers (or click "Connect" from Object Explorer), you get a dialog box where you can enter SA credentials (or whatever) to log into the server, manage the databases, run queries, etc.

But the actual object explorer window doesn't care about those sa credentials, it is using your local machine credentials.

So, you (USER A) open up SSMS and hit "Connect" in Object Explorer and fill in information (sysadmin) to connect to SERVER A - if USER A is a local administrator on SERVER A, you will get the green arrow and be able to manage the Agent by right-clicking in Object Explorer. If USER A is not a local admin on SERVER A, you won't be able to. It makes no difference at all that you logged into SSMS (connected to SERVER A through SSMS) as sysadmin.

I hope that helps, it sounds confusing when I read it, but hopefully I made it clear enough.