Question

Are you allowed to locally store tweets obtained using the twitter api?
http://apiwiki.twitter.com/Twitter-Search-API-Method%3A-search

Answer 1

From the Twitter TOS:

By submitting, posting or displaying
Content on or through the Services,
you grant us a worldwide,
non-exclusive, royalty-free license
(with the right to sublicense) to use,
copy, reproduce, process, adapt,
modify, publish, transmit, display and
distribute such Content in any and all
media or distribution methods (now
known or later developed).

[TIP] This license is you authorizing
us to make your Tweets available to
the rest of the world and to let
others do the same. But what’s yours
is yours – you own your content.

In other words, yes, you can store them, and reproduce them at will. That page also links to
the API rules, which have a few requirements, like attributing tweets, not editing them to change their meaning, and getting permission before using them commercially (like on t-shirts).