The Reform position is well-known and since I do not subscribe to that position, I will leave it to others to address it.
Conservative Judaism, historically, has held positions which are closely in line with the traditional halakhic position, that one is Jewish if born of a Jewish mother or duly converted by a Beit Din—Rabbinical Court.
How all of this applies varies and it appears that regionally even within orthodoxy there are varying standards.
Nonetheless, traditional Judaism holds that there can be no marriage between Jews and non-Jews.
It is not a question of how much the couple loves one another, or how wonderful they are. The halakhic answer is, no. Now, if the non-Jew is truly sincere in their wish to become Jewish through halakhic means, then it would appear that they may marry the Jewish partner.
But even this is not so straightforward. I would be less than forthcoming if I did not point out that there have been and are religious communities that frown upon conversion for marriage purposes, even if there is some real sincerity in the desire to be part of Judaism.
In America we speak of intermarriage, meaning a Jew marrying another of a non-Jewish origin. In other countries, such as Australia, Jews do not speak of “intermarriage”, rather “marrying out.” It appears to me that in America we emphasize the joining in, whereas in other countries the emphasis is on the loss of another Jew from the Jewish fold.
In fact, I think that it can be borne out statistically that more than the Jew brings in another to the Jewish fold; a Jew is lost to Judaism altogether when they marry a non-Jew. The Jew generally joins with the majority population, rather than the non-Jew becoming “Jewish.”
When asking about a couple intending to raise their children as Jewish, the fact remains halakhically, that any child born to a Jewish mother is Jewish by definition.
Any child born to a non-Jewish mother is not Jewish by definition, period. If the child upon reaching maturity desires of their own volition to convert, they may convert, just as any other person could, but the fact that the couple decided to intermarry and raise the child as Jewish, does not ipso facto make the child Jewish, from an halakhic point of view.
I know that this is not the prevailing position in Reform Judaism. As a matter of fact, I personally, had been asked to participate in the conversion of an adopted child of a Reform Jewish couple (both Jews), where their Reform rabbi would not convert the child, feeling that the non-Jewish child was raised within the “temple” and did not require any further steps to ensure Jewishness.
The couple did not follow their rabbi’s direction and sought an halakhic conversion, by a duly constituted Beit Din—Rabbinical Court.
On another occasion, I was asked to sit on a Beit Din converting a natural born child of an intermarried couple, where the mother was non-Jewish, but the couple had previously agreed to raise any children born of their union as Jews.
I declined to sit on that Beit Din, as I feel that if the child desires to become Jewish when they reach their majority, they can at that time choose to become Jewish.
As I stated at the outset, this position is in accordance with the standards of halakhic practice and may not be popular amongst non-halakhic segments of the Jewish population, but that is the historic Jewish position within Rabbinic Judaism.