Before my "tradversion", I would have said that if the other party appeared to be on the path to conversion, then maybe it would be okay to date non-Catholics. But being more familiar with the mind of the Church on the question now, I realise it's actually a big no-no.
As others have said, you should never start keeping company with someone if they're not what you want in a spouse in the here and now. It's most imprudent to carry on dating and being engaged for very long (I'd say, a year from starting to court to getting married), so it's keeping yourself in an unnecessary occasion of sin to prolong courting while you wait around for the person to convert before you can marry. By all means, keep your eye on someone from a distance and snap them up if/when they do convert (I know a guy who did this. How romantic!). But I don't think it's appropriate to date them.
I also think it would be imprudent, possibly sinful, to put yourself in the position of dating someone whose views on chastity aren't in line with traditional Catholic thinking (this sermon helped clarify that for me: http://www.audiosancto.org/sermon/20060806-Occasions-of-Sin-and-Company-Keeping.html
). Forget no extramarital relations; how many young people are going to understand no passionate kissing, little time alone together, etc. They may agree to those things because they like you, but when push comes to shove it's unlikely many would uphold moral beliefs they themselves don't prescribe to. You definately want to be helping each other, not dragging each other down.
I don't feel I've expressed myself very clearly but I thought I'd throw in some authoritative quotes about the evils of mixed marriages to help clarify how you wouldn't want to end up deeply emotionally attached to someone who you shouldn't (even couldn't) actually marry:
Regarding mixed marriages between Catholics and non-Catholics [i.e., valid, but illicit without the required conditions —Ed.], Pope Clement XI wrote, "The Church abhors such marriages, which greatly endanger and deform the spirit" (June 25, 1706).
In Matrimonia Quae in Locis (Nov. 4, 1741) Benedict XIV spoke of "sacrilegious unions," of "detestable marriages, always…condemned and prohibited by Holy Mother Church." And Pope Pius VI wrote, "This type of conduct gives rise to the danger of perverting the Catholic party…We cannot abandon our position because we do not have the right to do so" (Letter to the Archbishop of Malines, July, 1782)....
The Church has always abhorred marriages between Catholics and heretics and has always prohibited them with very severe laws because they always conceal the grave danger of perversion and estrangement from the faith of the Catholic spouse, and because the Catholic education of the children of both sexes is always doubtful and suspect. (Letter to the Archbishop of Mayence, Oct. 8, 1803)The Church has always abhorred marriages between Catholics and heretics and has always prohibited them with very severe laws because they always conceal the grave danger of perversion and estrangement from the faith of the Catholic spouse, and because the Catholic education of the children of both sexes is always doubtful and suspect. ([Pope Pius VII], Letter to the Archbishop of Mayence, Oct. 8, 1803)
In no way can tolerance [of such marriages] be seen as their being approved and allowed. Rather, such marriages are only tolerated out of the need to avoid even greater evils, and permission is not given wholeheartedly.http://www.sspx.org/miscellaneous/okaying_mixed_marriages.htm
The Church Fathers had already recognized that in such marriages there is no image of Christ’s union with His Church, but rather an image of Christ’s own being prostituted. (See Tertullian, Ad uxorem, 2, 3-4; St. Ambrose, Ep. 19,7; St. Jerome, Adv.Lovin. 1,10; St. Augustine, De fide et oper. 19,38).
And yes, I have known mixed marriages where one person converted. From my experience, often the Catholic party wasn't all that fervent to start with and the converted party ends up more devout. I wouldn't like to think what state the Catholic parties would be in if their wives hadn't converted. My sister solemnized a mixed marriage before our return to tradition and it's not a situation I would recommend putting yourself in. I'm soooo glad I married a good Catholic! I know it sounds cliche, but marriage is hard enough without throwing that spanner in the works. God makes you one person, yes, but you still have to work at being of one heart and one mind.