I've toured with drops and trekking bars. My preference is..... both of them. It all depends on the getting the right setup for you. Personally, I think one of the biggest issues with drops is installing them too low with respect to the saddle height. Likewise, I almost gave up on my trekking bars because I just couldn't find the right setup -I eventually stumbled on it by accident when I took the trekking bars off my touring bike to put on my backup touring bike, and like a bursting flash of light they were suddenly very comfortable!
The things I do find are that trekking bars are easier for standing out of the pedals, and that for shifting, unless I use STIs, mtb shifters are closer to hand and more convenient (at least for me). Drops are nice for getting more aerodynamic when you have that nasty headwind. Anyway, assuming you can get comfortable with either setup, here are some thoughts:
Trekking bars can accommodate mtb shifters, levers, etc. I recommend trekking bars for anyone who is initially converting an mtb over to a commuter or tourer based on the fact all you need is the bars, most likely a new stem, and perhaps a little bar tape. Factor in the fact that the ubiquitous v-brakes can be easily used with standard long pull mtb levers and it makes for fast, easy and cheap conversions -even if you need to buy new stuff, usually you can get mtb stuff for cheap.
Drop bars soon rack up the expense *if* you have no existing componentry. If you have v-brakes then you are more limited with your braking choices, particularly if you choose to go the STI/Ergo route as you'll need some adaptor thingimajigs (as well as a road specific front derailleur). Assuming you have cantis, then standard short pull road bar levers can be used. One the biggest expenses is with the shifters -STIs/Ergos/bar ends aren't cheap, particularly compared to mtb Ebay shifter options (e.g. I use $10 9-speed SRAM Centera shifters I bought from Ebay on my trekking bars which I find are fine). Another alternative is to use downtube shifters, but personally I like shifting controls close to hand (yet another alternative is to use downtube shifters with Kelly Takeoffs, my own setup I use and love them -though they may not work with the Randonneurs. Also, you could try barend shifters with Pauls thumbies).
If you are short of money or have existing mtb components, I'd say try the trekking bars first as it would be a relatively low cost experiment to see if you like them. If you have the money, well, just make a choice based on what you think you'd prefer!