twitter.com requires https access in order to publish tweets. Otherwise, this message appears in jenkins.log:

***

hudson.plugins.twitter.messages.AsyncTweetDeliverer$1 onException
WARNING: Exception updating Twitter status: 403:The request is understood, but it has been refused. An accompanying error message
will explain why. This code is used when requests are being denied due to update limits (https://support.twitter.com/articles/1536
4-about-twitter-limits-update-api-dm-and-following).
message - SSL is required
code - 92

Relevant discussions can be found on the Internet at:
http://www.google.co.jp/search?q=68d96c5d or
http://www.google.co.jp/search?q=f7f78983
TwitterException

{exceptionCode=[68d96c5d-f7f78983], statusCode=403, message=SSL is required, code=92, retryAfter=-1, rateLimitStat us=null, version=3.0.3}

***

The released version of the twitter-plugin, 0.7, uses twitter4j-async-3.0.3.jar.
Only twitter4j-async-3.0.5.jar and later use https. So tweets aren't published at all.

A workaround was to replace the file:
~/plugins/twitter/WEB-INF/lib/twitter4j-async-3.0.3.jar with a downloaded twitter4j-async-3.0.5.jar

I have to take a shower after workarounds like this one. But it works.