Is there any downside for using a leading double slash to inherit the protocol in a URL? i.e. src="//domain.com"

If the browser supports RFC 1808 Section 4, RFC 2396 Section 5.2, or RFC 3986 Section 5.2, then it will indeed use the page URL's scheme for references that begin with "//".

When used on a link or @import, IE7/IE8 will download the file twice per http://paulirish.com/2010/the-protocol-relative-url/

Update from 2014:

Now that SSL is encouraged for everyone and doesn’t have performance concerns, this technique is now an anti-pattern. If the asset you need is available on SSL, then always use the https:// asset.

One downside occurs if your URLs are viewed outside the context of a web page. For example, an email message sitting in an email client (say, Outlook) effectively has no URL, and when you're viewing a message containing a protocol-relative URL, there is no obvious protocol context at all (the message itself is independent of the protocol used to fetch it, whether it's POP3, IMAP, Exchange, uucp or whatever) so the URL has no protocol to be relative to. I've not investigated compatibility with email clients to see what they do when presented with a missing protocol handler - I'm guessing that most will take a guess at http. Apple Mail refuses to let you enter a URL without a protocol. It's analogous to the way that relative URLs do not work in email because of a similarly missing context.

Similar problems could occur in other non-HTTP contexts such as in tweets, SMS messages, Word documents etc.

The more general explanation is that anonymous protocol URLs cannot work in isolation; there must be a relevant context. In a typical web page it's thus fine to pull in a script library that way, but any external links should always specify a protocol. I did try one simple test: //stackoverflow.com maps to file:///stackoverflow.com in all browsers I tried it in, so they really don't work by themselves.