The DropBox streaming player sucks butt (shokes, stutters, takes forever to load... if it works at all). For sending large files for them to download and watch locally (on their own computers) it's fine.
However to just send files like that all they need is a Gmail account which gives access to GDrive which IIRC provides 15gb of online storage which can be shared. I have not tried GDrive yet though so not sure how good it is. Microsoft's "Live" has introduced OneDrive as well which is similar but IMO sucks compared to DropBox (it has given me a very hard time trying to retrieve files that have been sent to me).
Youtube is probably the most stable player out there for this stuff. I have yet to create an account so I'm not sure if it allows for you to mark vids as Private that can be shared with others but it seems like the type of feature they would have.
There are TONS of decent video storage players that the "pirates" use to upload shows and stuff but that's a little skeevy. A good example of such a player would be the old (and now defunct) Megavideo but that's not something you really want for personal stuff methinks.
With the inception of the new HTML5 inline vid player capability they could set up a simple web server locally or get a GoDaddy account or something and use their alotted space to host the vids. Then as long as the viewer has a new, HTML5 capable browser they could just stream the vids directly from there. This would require setting up a simple website and coding in the vids but it's not hard (there are templates and the code can be copy/pasta'd). This of course means the site is likely viewable by the entire intertubes (not sure if there are ways to hide a web address) but they could just register a really long and random alphanumeric domain name that is unlikely to pop up in any search engines or draw any attention. In fact you could make it a login site or perhaps set it up so the intended recipients have admin priveleges or something to viewed secure areas (which is where you'd store the vids).
Meh... just taking a break so thought I'd toss out some meandering ramblings.
Cheers.