First, there's one use of 「恥じる」 you may have not been acquainted with.
From 広辞苑 under 「恥じる」:
③(多く、否定を伴う)ひけをとる。劣る。 【(often accompanied by a negative) compare unfavorably with; to be inferior】
As hinted at in the dictionary, in this use the word often appears in the construction 「(~に)恥じない」, which means: "be not inferior to; i.e. be comparable with, be as good as, or measure up to someone/something". Thus:
「あいつに恥じない成果」 = "results that are as good as hers/his"
but
「人に恥じることはなにもない。」= "There's nothing to be ashamed of (towards people)."
Notice in the latter it's 「に恥じる」, not 「に恥じない」, so that should be a good indicator that it's not used in the sense I mentioned above.
As for your questions:
恥じる: To feel ashamed.
僕が恥じる:I feel ashamed.
僕に恥じる:To feel ashamed towards me.
Is it correct?
Correct.
Is it there a way to understand this?
I have nothing definitive or even terribly pertinent to say about this, but to take your examples:
There is something to feel ashamed in laughing at people. → 人を笑うということには、どこか恥じるべきところがある。(私なりの訳)
In this particular sentence I feel more comfortable with 「には」 instead of just 「に」, but using 「に」 would be perfectly grammatical and understandable. Personal preference, maybe.
I feel ashamed about that towards her. → そのことについて、私は彼女に恥じています。(私なりの訳)
In the "towards" or "against" sense, more often than not 「に」 can be replaced with 「に対して」without changing the meaning. Here, 「私は彼女に対して恥じています」 would be as acceptable and understandable.
