I don't believe this is possible - there's just too much going on with stacks and different block types to be able to directly share stack contents via some trick with the databases.

You may be able to come up with some sneaky ways to fetch the data from one section of a site to display on another, but those ways are likely to involve custom coding. You could perhaps on one site create a blank page and put on the stack, and then on the other site create a custom block to go fetch that page and strip off the stuff you don't need. A bit hacky though!

If you're just talking about page lists, you could try utilising the RSS feed on the source site and the 'RSS Displayer' block on the other, but that's really just an individual block and not a full stack.

You could pull content from the page at one location into another location. Doing so may have issues with any css or JavaScript assets, but for straight forward html page areas it will work.

Have a look at the Box Grabber addon and my own Universal Content Puller.