Who are the illegal Jews who have come to Israel under the Law of Return?

When the Jewish state was established, one of the very first laws promulgated by the Knesset was the Law of Return.

New Olim from Ukraine, December 24th, 2018. (photo credit: OLIVIA FITUSSI)
New Olim from Ukraine, December 24th, 2018.
(photo credit: OLIVIA FITUSSI)
A headline in the Jerusalem Post (January 3, 2019) informed us that a “Majority of 2018’s immigrants are not legally Jewish,” and the article went on to state that “there are approximately 400,000 Israelis, mostly from the former Soviet Union, who immigrated to Israel under the Law of Return but who are not Jewish, and this number is growing every year, due to both natural growth and continued immigration.”
The article continues, suggesting that “one approach to solve the problem of Jewish intermarriage has been to push for a higher rate of conversion and to make conversion easier,” by citing an independent Orthodox system of Courts called Giyur K’Halacha established in 2015. And finally the article concludes by quoting the Chief Rabbi of Kiev who describes such solutions as “disastrous,” saying that by bringing such non-legal Jews into Israel “we, with our own hands, are facilitating intermarriage and assimilation in the State of Israel.” 
Permit me, as the chief justice (av bet din) of the first of the independent courts of Giyur K’Halacha, to give a fuller background as to why Giyur K’Halacha began to operate as well as into the true halachic status of those who were “not legally Jewish,” in the words of the headline cited above.
When the Jewish state was established, one of the very first laws promulgated by the Knesset was the Law of Return: After 2,000 years of Jewish homeless wanderings and persecutions, any Jews from anywhere on the globe seeking safe haven would be welcome to immigrate to Israel and gain automatic citizenship.
However, the next necessary question to be answered proved to be the most difficult one: Who is a Jew? Questionnaires were sent to Jewish philosophers and halachists throughout the world, and, as you conjecture, the responses were many and varied with no unanimity at all.
The decision was finally decided by Knesset on the basis of Jewish history and our worst Jewish exile experience; whoever was considered to have been Jewish enough to be sent to Auschwitz under the worst antisemites in history, the Nazi regime, would be Jewish enough for Israel’s Right of Return. And since any one with one Jewish grandparent, either paternal or maternal, was sent to Auschwitz, so anyone with one Jewish grandparent, either paternal or maternal, was eligible for the Right of Return and automatic citizenship in Israel.
Approximately 1,300,000 Russian Jews arrived in Israel after Perestroika opened the Iron Curtain in 1994, approximately 300,000 of them with Jewish paternal and not Jewish maternal background. This is what led to the alarming statistics cited above.
Allow me to clarify one critical halachic issue: yes, Jewish law rules that maternal  and not paternal parentage decides the religion of the child, but one who is born to a Jewish father is not considered a gentile (goy); great halachic decisors such as Rav Moshe Isserless (the Rema, 1530-1572) and Rav Moshe Margolies (the P’nei Moshe 1718-1780) call them “zera yisrael” (Jewish seed) – because, after all, the father does provide his progeny with DNA and dare not be completely discarded when discussing ancestry. Rav Haim Amsalem has published a two-volume work titled Zera Yisrael, citing the dominant view that although these “offspring” are not Jews according to Halacha, neither are they to be considered gentiles. Rav Ben-Zion Uziel, the Sephardi chief rabbi of Israel for more than three decades, has ruled that such Jewish seed must be embraced and encouraged to return home to their Jewish forbears, requiring only circumcision for males and ritual immersion for males and females. They are part of the “Ingathering of the Exiles” that our prophets have foreseen as accompanying our return to the Jewish state.
It was primarily for the sake of the offspring of these zera yisrael that Giyur K’Halacha was originally formed. Our goal is not to facilitate intermarriage (God forbid) or even to make conversion easier. We are merely attempting to put into practice the entire spectrum of Jewish Law in order to lovingly bring our Jewish exiles home to Judaism and to our Jewish state.
The writer is chief rabbi of Efrat and founder, chancellor emeritus and rosh hayeshiva of Ohr Torah Stone.