What could be an alternative word implying
Share this post to my network! for use in a social application ?
http://sharethis.com/ have used a short catchy phrase 'Share this' and used a really good icon indicating connected but diverging nodes in a network.
I'd suggest something similar. It depends if there is an intermediate stage between clicking the button and submitting the final share, but Share with, Share to, Share this are all short and to the point.
Of course maybe your social network could have a related 'action', like retweet works well for twitter.
I would consider shifting the emphasis to the word "post" ... "Post this to my network." I know that it might create ambiguity if you have already locked in on some other use of the word Post, but that may not be a problem.
"Syndicate this to my network" would be pretty out there but sure would grab the attention. "wow... I can syndicate something? hmm!!"
This probably depends on the kind of design you're making. SE itself just uses icons and lets people figure it out because the general audience knows how to share if it needs to.
If you can figure out the general web-savvy of your audience, you could define the right word pretty quickly.
As a rule, if space is not an issue, it's great to be specific: Share this on Facebook/Twitter works best.
I'd start by looking at other copy in the site and try to find terms that match the voice that's already set there. Is it colloquial, is it formal, is it technical or new age? Driving your search for terms from the brand while ensuring that it's still understandable will help the feature fit with the overall site, and you'll probably feel better about the label you use as well. It's a roundabout way to name something, but the legwork often turns up words you might not otherwise think of.