As a rule, yes, but they probably won't require any more bandwidth than a disaster recovery or a managed service backup would. They are going to require a lot bandwidth. The reason why people want to archive email and reduce email storage is because it is such a big user of capacity and capacity equals throughput. If you're sending around multi-megabyte files, all of that stuff is going to have to go over the wire to a hosted provider or a managed service. So that is of course a big concern, the amount of bandwidth you are going to need.
You can mitigate that in a number of ways; first, by going with your current hosting provider. In a lot of cases, you might have hosted email already and your provider might be able to do email archiving for you right there and the bandwidth would be negligible. Or, you may find that you have other applications and other connectivity options already in place that you can leverage with those hosting providers.
For the big guys, they're relying on just internet connectivity. So it's just a question of talking to your connectivity provider and dialing up your internet speed.
Finally, there are a lot options out there that are technical solutions for this problem, that apply capabilities like data deduplication and compression to reduce the amount of bandwidth required. So the real specialist applications for email archiving, like Zantaz, are going to have an onsite component and an offsite component. The onsite component is going to reduce your bandwidth significantly by applying principles like deduplication to it as a way to both reduce your bandwidth requirements and to accelerate the service.
So they do some caching and acceleration, so it's not always going out over the internet to get everything. There are a lot of ways that you can mitigate it. You can look at trying to keep it local or near your email server; you can try to go with just a generic provider and bulk up on the bandwidth. Or you can try to get something that is really tuned to mitigate the bandwidth problem.
Check out the entire Email Archiving FAQ.