As it happens, my last trip to Rome, a couple of years ago, I stopped by the Great Synagogue for the evening services, and was halted outside the fence by armed guards, who wouldn't let me in until I had spoken with one of the known members of the synagogue (in Hebrew, since he didn't know English and I didn't know Italian) and had convinced him I was a Jew who just wanted to pray. Why the fear? They had been persistently vandalized and their members harrassed by local anti-Semites, mostly in the guise of Christian anti-Judaism....
wow...the Synagogue is for Jews. Those guys just wanted to be sure you were a Jew. what's wrong with that?
In fact I've never wanted to enter there because I'm not Jewish