With 10% of the Roman Empire being Jewish (perhaps 25% in the east), with Roman Legions moving from one end of the Empire to the next, with Jews intimately involved in the Empire (at war with it, granted privileges by it and doing business with it all depending on the time frame), many if not the majority of people of European descent probably have Jewish ancestors.
I would not doubt that either.
Indeed, the mixing must have gone in both directions, and I have no doubt that many Christian Italian, French, Czech, and German people have partial Israelite ancestry.
A recent genetic study suggested Spanish and Portuguese people have a substantial portion of Israelite ancestry - 10 to 30 percent.
As for converts to Judaism, a new book some of you may be interested in is Eric Maroney’s “The Other Zions: The Lost Histories of Jewish Nations”.
http://www.amazon.com/exec/obidos/ASIN/1442200456/
Maroney is a fine writer who explores conversions to Judaism in Adiabene, Yemen, and elsewhere and how some of those converts had high status and political power.
Of related interest is my book “The Jews of Khazaria”, about the Khazar people’s conversion to Judaism and the issue of partial Slavic and Khazar ancestry in East European Jews. Although these new DNA studies showed those elements to be less significant than partial Italian/Greek/Mediterranean ancestry, they still exist. R1a1 was found in about 7.5% of East European Jews in Dr. Behar’s study “Contrasting patterns of Y chromosome variation in Ashkenazi Jewish and host non-Jewish European populations”.
Some East European Jews had Slavic names and spoke Slavic before the influx of Yiddish-speaking Jews. They could very well have intermarried with the local Belarusians and Ukrainians.
http://www.khazaria.com/brook.html