What you want is available now in Android. For example, I currently have 3 office suites installed on my Transformer tablet to test which one would be best as a daily driver. So, when I open Dropbox and click on some Word doc, there will be a pop-up window asking me which one of the 3 word processors should open it. I can get the same sort of thing with any image files, text files, etc: options to open the file with whatever app can handle it.
Sending/sharing files works similarly. Go to send an image file and you'll get the option to attach it to an email, send it via twitter, put it in Facebook, send it via mms (if your device is a phone), put it in Dropbox, incorporate it into a note in Evernote, etc. Third party programs will add the options to send it to network shares or ftp.
Android lets you have access to your filesystem from the tablet / phone itself, so you can have that level of control if you want it. Most tablets / phones come with a very basic file manager and more advanced ones are a quick download away. If you would prefer to do everything from a command line shell, this is also available.
Android doesn't lock down the user the way an iPad does.
That's not what I'm looking for. Your data is still owned by an app, in your case Dropbox (from the perspective of your tablet experience). Note that you must first go into an app (Dropbox) and push it to another app. You can't do the opposite, you can't pull. You can't, from Dropbox, pull in a document created in App X. If you're lucky App X has a "share" feature that you can send to Dropbox but that is not guaranteed.
A lot of people think the file system metaphor is a bad one, but I can't think of an alternative that keeps the user in control of their own data.
Sending/sharing files works similarly. Go to send an image file and you'll get the option to attach it to an email, send it via twitter, put it in Facebook, send it via mms (if your device is a phone), put it in Dropbox, incorporate it into a note in Evernote, etc. Third party programs will add the options to send it to network shares or ftp.
Android lets you have access to your filesystem from the tablet / phone itself, so you can have that level of control if you want it. Most tablets / phones come with a very basic file manager and more advanced ones are a quick download away. If you would prefer to do everything from a command line shell, this is also available.
Android doesn't lock down the user the way an iPad does.