Hiya,
I've looked at JamRoom in the past, and have decided to use Dolphin instead. I suppose the real question that you need to ask yourself firstly is which features do you want to use from JamRoom and which features do you want to use from Dolphin.
You could hire a developer to create a bridge, this would share login details or give single-sign-on, but having the two scripts work in tandem would require much more than this.
You need to decide which scripts user database, messaging, forums, profiles, media etc would be used - and change the other script to point at it.
You'd need to do the same with any other modules. It could get very hairy for a programmer!
Once all that was done, you'd then have to deal with the aesthetics - getting someone to make the look and feel match between the two scripts.
It's a big task... It may be better using one script and getting a programmer to custom write modules you need for whichever script you decide to use?
If you can list out the features you want from the two scripts, you'll get a much better idea of which way to play this.
One thing to remember is that Dolphin by itself is free and GPL'd; JamRoom is at cost and closed source.
Personally I like the chart features of JamRoom, and fan pages - but D7 to me offers a lot more... It shouldn't take a competent programmer long to create add-ons for D7 to emulate these features.... And certainly a lot less time and money than trying to bridge and force the two scripts to work with each other.
On the flip-side, you've already made a large commitment in time and money in setting up your current sites on JamRoom.... You need to look at whether you want to go through the process of migrating everything to a new site, and going through the setup process again. Also consider likely costs to get custom programming for JamRoom (if indeed you can, I haven't looked) to get the modules of Dolphin you like added to it.
You also need to consider the likely costs for say the next few years for upgrades of both systems.
The other major problem with making Dolhin and JamRoom work together is when it does come to upgrade cycles for both applications, you'll need to get a programmer back in to make things continue to work - and as JamRoom and Dolhin are different products and have different release cycles, this could continue to prove very expensive.
Sorry for the essay... And hope I've helped. I don't mean to put a downer on your plans - and you can get a site operating in the way you want to, I just don't think trying to make the two scripts in this case work together is the best solution for you or your users.
Personally - if I was you - I'd go the Dolphin route - being GPL there's less restrictions on what you can do, and more likely to be programmers with knowledge of the script (or willing to invest time to get to grips with the script). You can then get the features you want written for Dolphin coded up and implemented.
Iain.