The reason I know what I know about Jews for Jesus is this: I encountered them while converting from Christianity to Judaism. I was not born Jewish, have no Jewish relatives that I know of (all staunch Protestants, thank you very much), and no particularly strong exposure to Jewish culture. The story of why I converted is, literally, another story, too long to print here, I think.
It was while I was studying for my conversion that they came to our synagogue over the course of three consecutive daytime Sabbath services, apparently as part of a campaign.
Don't get me wrong, it was not like these guys marched into the synagogue carrying crosses and singing "Onward Christian Soldiers". In total, only two of them ever came to our synagogue. And in fairness, they were neither loud nor obnoxious, although on the last occasion, when both of them showed up, they seemed a bit, well, desparate.
How did I know they were from JfJ? As I said, I asked. As far as I know, I was the only one who did so directly, and I asked them point-blank. I knew about Jews for Jesus because Christian members of my family kept trying to get me to talk to these guys during and after my conversion (they have finally given up).
Their methods were "disruptive" in the sense that they took advantage of the fact that any Jew may speak freely in synagogue, even if doing so during actual services is considered somewhat rude.
In the case of the JfJ evangelists, they did not interrupt the prayers, but rather made interjections during the part of the service where the rabbi would give his "sermon", discussing the day's reading of the Torah and its significance. During this portion of the service the JfJ guys, dutifully dressed up in their tallitot and kippot, would occasionally ask leading questions.
Sometimes the questions were subtle, and sometimes they were not so subtle, like "so doesn't that prove that Jesus was the Messiah?", and so forth. After the services, the rabbi would gather the Judaism students (myself included), to discuss aspects of Judaism that were laid out in our course curriculum (literally called Judaism 101). This was when one of the guys really got annoying, almost hijacking the discussion. After three lessons affected this way, the rabbi finally took the guy aside for a private discussion and, as best I can tell, must have asked him to leave, because I never saw him again.
I was a Christian for 25 years. When I accepted Jesus as my personal savior at age 15, I felt the unmistakable wash of joy us Christians always talked about. But I never truly felt connected to God Himself. And God knows, I tried. Oh how I tried to understand and make sense of it, and believe! But the connection just didn't get made. I was not in touch with God.
To describe all the circumstances surrounding my conversion and the reasons for it would take up a rather large post unto itself. Suffice it to say that for me, Judaism is the way to connect with God, and Christianity is not. At age 40, my prayers were finally, truly answered.
Don't get me wrong, I gained many insights as a Christian, and don't regret my time as a Christian at all. But it took understanding God from the perspective of Judaism, and as a son of the covenant between God and Israel, for me to finally hear His voice. Now when I talk to God, he really answers! Still amazes me. frankly.
But don't mistake my testimony as claming that Judaism is the "One True Faith". Ask any Jew. It's not. It's the one true faith for Jews, that's all.
If Christianity works for you, and leads you to God, then it's your one true faith. And that's why we Jews don't come into Christian churches, ring your doorbells or call you on the phone to come convert to Judaism. It is our job to be a "light unto the nations", not to make them all Jewish.