First, even if you "installed an application to your dropbox", it would still be running from your local machine - that's just how Dropbox works.
Second, even if it were running the application from Dropbox's server(s) (it seems like that's what you mean?), it would still have to read from their servers - you can't magically eliminate a bottleneck just by moving it to a different physical computer; data still has to be read from disk.
Though a big datacenter seems like the perfect place to spread the bits around so that you can read from multiple physical disks/computers simultaneously to improve performance. This is unlike your laptop where you are probably not carrying multi-disk arrays around with you.
Second, even if it were running the application from Dropbox's server(s) (it seems like that's what you mean?), it would still have to read from their servers - you can't magically eliminate a bottleneck just by moving it to a different physical computer; data still has to be read from disk.