They started to piece the story together from hearsay.

他们开始根据传闻把事情的经过一点点拼湊起来。

She discovered a world of parties and pleasure she had hitherto only known by hearsay.

她发现了一个以前只是略有听闻的花花世界。

You are only supposing this on hearsay,you have no proof.

你只是根据传闻想象而已,并没有证据。