OK, there are a couple different things going on here.
Potentially inflammatory posts
It's certainly true that chat bystanders can often tell when a conversation is about to turn nasty. And you're definitely right that it's better to prevent nasty exchanges from breaking out in the first place than to allow them to happen and deal with them afterwards.
The thing is, you don't need moderators to do this!
In any community, when someone starts behaving inappropriately, the first and best way to stop that behavior is for other people in the community to come right out and call them on it. A simple "dude, don't say that" or "I'd rather you not talk about [whatever]" or "let's drop this for now" or something like that is often all it takes. If different people disagree on what's appropriate, the whole group can get together and discuss it, and come to some consensus. There's no need for moderators or any higher authority to get involved, and that's good - a community like our chat room should be able to maintain itself without having a moderator constantly watching over everyone's shoulder.
Actually inflammatory posts
Actually talking to people works when everyone is being reasonable and civil. But that's not always the case in practice. People's tempers flare up sometimes and their control can slip.
If you ask someone to drop an issue or tell them something they said is inappropriate, sometimes they respond by insulting you or spouting profanity and completely ignoring what you're trying to tell them. When calling someone out on their inappropriate posts doesn't work, and they escalate to "conversational violence" (insults, rudeness, etc.), that's what flagging is for. Flagging brings a post to the attention of the chat moderators, and possibly also 10k-rep users, from across the SE network.1 Sometimes a forceful word from a moderator is all it takes to get the discussion back on track, but if not, moderators (or a consensus of enough 10k-rep users) have the power to remove chatters who don't have the sense to remove themselves from an argument.
Here's the thing, though: flags show up to a lot of people. Unlike on the site, where we only have 4 moderators and a few dozen 10k-rep users, there are over 300 chat moderators and probably thousands of people with 10k reputation. Obviously not everyone is online at any given time, but still, that's a lot of people seeing a flag. So flags should be used with care. You shouldn't be casting flags unless things have really escalated so badly that the people in the room can't deal with it anymore. (Or for spam, but that rarely happens.)
The typical outcome of a flag is that the flagged message will be deleted and the posting user suspended for 30 minutes (or longer if circumstances warrant). For anything that doesn't deserve that kind of a response... well, is it really so bad you (people in the room) can't handle it yourself? Probably not.
You'll notice that nowhere in here did I address the main point of the question, which was how to contact only the site mods privately. Honestly, that's probably not that useful. If we (mods) are logged in at the time, we're probably watching the chat room anyway, and we'll step in when things start to get out of hand. Otherwise, we're not going to see any message you might send us until we do log in, by which time it will be too late to defuse the situation.
The fact is, chat moves fast, and blowups in chat should likewise be handled fast. So once things get to the point where you need to call in outside help (and remember, they should not get to that point easily), it makes sense for that call to reach someone who can respond quickly. If one of us Physics mods isn't already in the room, it's better to bring in a chat mod from another site rather than waiting around for one of us to get there.
1As Shog9 pointed out, there are two kinds of flags. The inappropriate/offensive flag goes to mods and 10k-rep users, but there is also a custom flag that is only seen by mods. The situations described in this answer will typically only need a generic inappropriate/offensive flag, if any. See more information about chat flags on Meta Stack Exchange.