I’ll simply add that “Messianic Judaism”, “Jews for Jesus”, etc. (there are lots of groups/sects and no two are alike) are not practicing Judaism but a form of Christianity.
A Jew who “accepts Christ” has crossed the boundaries into heresy/apostasy. Belief in, or acceptance of, Jesus as the Messiah/Son of God/person in a triune God/Incarnation of God/ is completely, totally & utterly incompatible with the beliefs of traditional, normative Judaism, and always has been. This circle can never be squared; “Jews for Jesus” and the various “Messianic Jewish” groups, et. al. will always be unnatural jerry-rigged hybrids. One can believe in Judaism or Christianity but not both at the same time, it’s one or the other. I certainly believe in honest & friendly (always!) dialogue between Jews and Christians, I also believe that such dialogue must be based on a recognition that ours are two separate faiths.
A Jew is anyone whose birthmother was a Jew or who underwent an orthodox conversion. Thus, while many/most/some(?) so-called “Messianic Jews” may certainly be Jews, what they believe and practice is certainly not Judaism.