One point is missing from the explanations above, the best way to add your testers into the SCRUM process is by making sure they are part of the scrum team and work together with the rest of the team (devs, PO, etc) in the Sprint. Most of the time this is not really done, and all you end up having is (in the best case) a Mini-Waterfall process.
Now let me explain. There is little to add to the extensive hardware and environment explanations above, you can work with staged servers, or even better make it an internal feature to have the scripts in place that will allow testers to create their own environments when they want to (if you are using any CI framework chances are you already have all the parts needed in there).
What is bothering me is that you said that your testers "have their 'own' sprint".
The main problem that I've seen when getting testers involved into the SCRUM process is that they are not really part of the process itself. Sometimes the feeling is that they are not technical enough to work really close to developers, other times developers simply don't want to be bothered by explaining to testers what they are doing (until they are finished - not done!), other times it is simply a case of management not explaining that this is what is expected from the team.
In a nutshell, each User Story should have a technical owner and a testing owner. They should work together all the time and testing should start as soon as possible, even as short "informal clean-up tests" in the developers environment. After all the idea is to cut the Red Tape by eliminating all the unnecessary bureaucracy in the process.
Testers should also explain to developers the kind of testing they should be doing before telling the QA they can have a go on the feature. Manual testing is as much the responsibility of the developer as it is of the tester.
In short, if you want to have testers as part of your development, even more important than having the right infrastructure in place, you need to have the right mind-set in place, and this means changing the rules of the game and in many cases the way each person in the team sees his task and responsibility.
I wrote a couple of post on the subject in my blog, in case I didn't bother you too much up to now, you may find these interesting.
Switching to Agile, not as simple as changing your T-Shirt
Agile Thinking instead of Agile Testing