Way way back when, Middle Ages era, rabbis debated whether Christianity was “idolatrous” on account of having three gods and having lots of idols/icons. I don’t want to get into the argument of whether Christianity does in fact have three gods, I just want to describe how it looks to an outsider. When you see people praying to a statue of the mother of one of the gods, it looks amazingly like ancient paganism.
The discussion was important, because Jews are supposed to accept martyrdom rather than to worship idols; hence, if faced with forced conversion to Christianity, what should be done? If Christianity was idolatrous, then it was better to accept martyrdom. If Christianity was not idolatrous, then perhaps forced conversion was preferable to death.
The issue was finally resolved (after considerable nastiness and objection on the part of the Christian leadership) that Christianity was not idolatrous. However, Judaism does not have a central authority, so different groups, branches, sects, and individuals were free to interpret how they wished. In areas where there was a high level of persecution by Christians, Jews tended to react by separating themselves even further from Christians, and held to the notion that Christianity was idolatry.
For such people and in such areas, it was forbidden to enter a Christian church. Many Jews would cross the street so as not to walk alongside a Christian Church, and even (surruptiously) spit a curse. It’s not a surprising reaction when the Christian leadership sponsored pogroms (outbreaks of violence, murder and theft against Jews) and similar persecutions.
For modern times, and for present day America, where persecutions are no longer legal, and are not sanctioned by Christian churches (well, except in rare circs), there has been more coming together. Most modern Jews do not have a problem with joint (neutral) worship services, or with shared experiences. However, there is usually still a line somewhere.
Jews who are educated in their religion know that they are not supposed to bow to idols, nor to give the appearance of such. Hence, their behavior in a church might be somewhat awkward, since they could not give the appearance of bowing towards statues. Leaning down to tie your shoe in front of a statue of a god might be seen as idol worship. In the example Robyn gives of shared space, for instance, Jews using a church usually cover the statues and pictures.
Pesonal anecdote: When I was in the boy scouts, in the mid 1950s, our troop went to an Easter service as a “group religious activity.” I have no idea what Church it was, but I was the only Jews, and I have to tell you that it was pretty awful listening to the pastor rant and rave about how the Jews did this and the Jews did that and the Jews killed Jesus, and to see all the people crying. I decided I’d never go to a Christian service again.
Later in life, in college, I became good friends with a Lutheran pastor, and we’ve visited them and attended services, usually every couple of years, and it’s a whole different experience. I’m not sure how much of the difference is Easter vs non-Easter, how much is the denomination, how much is him vs that fire-and-brimstone preacher, and how much is the year 2003 vs the 1950s. Whatever it is, it’s certainly a more positive trend towards mutual understanding, and movement away from the hatreds and bigotries of prior ages.