Sending a database dump
Of course you can put it up on Dropbox - or anywhere else. 

Depends on how secure you want your data to be? Do you want the NSA looking at it? Dropbox employees? 

I would put up an encrypted version of your mysqldump and securely transmit the key to your colleague - either face to face or over encrypted email, or have the password/key sent by courrier. 

It all depends on how much the data is worth to you and/or your company. Is it worth a courrier? Do I encrypt my email? Those are questions only you can answer. If your internal system is good, then go with that.

The key (no pun intended) question is "how much is your data worth (now and in the future)"? You should afford your data as much protection as is reasonable given its value to you/your organisation. 

If you're interested in these matters, go to the source of all wisdom - Bruce Schneier. "Security is a process, not a state". Once you start thinking in these terms, then solutions will become apparent. What is your data worth now? In a day's time? A year's? What would be the result if all of this data went live on internet now? If you wish to learn more, then I suggest you go to the crypto group - you ain't seen paranoid until you've lurked there for a while! :-)