I recently received a lengthy correction from a rabbinic colleague that a more correct usage would l'shalom. With his permission, the explanation follows.
The form "be-shalom" first appears in Genesis 15:15 "thou shalt go to thy forefathers in peace [be- shalom]." In 2 Samuel 3:21-23, Abner is thrice described as leaving David be-shalom. He is promptly murdered by Joab. In 2 Samuel 9, King David's last words to Absalom are: "lech be-shalom". (By strange coincidence, the Microsoft Word's Spellchecker changes the word "Be- shalom" to "Absalom"). Jeremiah (34:5) says "be-shalom you shall die." Thus, in liturgical usage, we say "be- shalom" in the phrase " lech be-shalom ve-tanuach al mishkavcha be- shalom" in the funeral service, and in the el maleh rah'amin.
In Modern Hebrew, one normally uses just " shalom", or the alternative "le- shalom" as: "Lech le-shalom" found in Jonathan's farewell to David (1 Samuel 20:42), and liturgically, in the Friday evening Shalom Aleichem: "bo'achem le-shalom...barechuni le- shalom...tzetchem le-shalom".
While it is not grammatically wrong to use "be-shalom " in all situations, the form is generally reserved for wishing a safe return ("tah'zor be- shalom" "tashuv be- shalom" (see e.g.: Genesis 28:21; Joshua 10:21; 2 Kings 22:17; 22:28) and for saying farewell to the dead.
The Talmud, at the very end of Brakhot (64a), therefore cautions us not to wish a person "be- shalom": "Haniftar me-h'avero al yomar lo lech be-shalom, elah lech le- shalom, sheharei yitro she-amar le-moshe lech le- shalom alah ve-hitzliah, david she-amar lo le-avshalom lech be- shalom halach ve-nitleh; ve-amar rabi avin ha-levi, ha-niftar min ha-met al yomar lo lech le- shalom elah lech be- shalom, she-ne'emar ve-ata tavo el avotekha be-shalom."
Update from Avinoam: due to my not being sufficiently clear, you lead your readers to imagine that signing off "le-shalom" would be acceptable. It would not. The accepted forms in written correspondence are: bivracha, biydidut, b'khavod rav. In speech, one says "shalom". Le-shalom is only used in specific collocations (like tzeitchem le-shPerhaps alom), and not at the end of letters or conversations.