Think about how the user will interpret your progress indicator.
If by standard progress indicator, you mean changing the cursor to a spinner or hourglass then I'd save that for system related tasks or where the application is effectively blocked from any further interaction (at any level).
The problem with using the cursor for this is that often this is seen more as a problem than as progress. The longer the cursor spins, the less confident the user is that it's going to come back.
Additionally, using the cursor does not contextualize the wait. Effectively you are saying the whole application is blocked.
In contrast, if you put an animated spinner right next to the message in your dialog, then you are contextualizing the wait in terms of the specific task going on in that dialog, and that is less worrying for the the user.
I would always try and locate the busy animation right where the update or next bit of information is going to appear - even if it's just a success message.
If by individual icon animation you mean an animation for example with two computers and symbols flying backwards and forwards between the two, or an animated signal moving along a network cable, or rings animated round the earth, then you don't need that kind of animation, so yes, save that for operating systems - it's too overkill inside an application dialog.
Just keep it simple - use the animated spinner. It's familiar, and does the job neatly and succinctly. It does not need to have any added value (baggage) by being an animation of information flowing about as this starts to delocalize the wait away from the context. The user doesn't care about information flying between computers - they care about the dialog and the response. By adding an animated internet/network, people will blame the internet for being slow, which leads to 'bigger picture frustration'.
By the way - I think perhaps 'Busy' is a better term than 'Progress' in this scenario.