Which IRC channel are you talking about? Anyways, yes, you can of course still run rippled (https://github.com/ripple/rippled) to directly connect to the P2P backend, you can also locally deploy the client that also powers rippletrade.com (https://github.com/ripple/ripple-client) if you don't want to get the JavaScript served from Ripple Labs' servers or you can use ripple-lib to write your own client (or interface directly with rippled or a different API endpoint like RippleREST).

The security model of rippletrade.com is similar to blockchain.info: Your private key gets encrypted locally with a username + password and stored (encrypted) at a third party server (aka. "blobvault"). Source code for all components is available under a free license, in case you are wondering.

You have to trust rippletrade.com to serve you the correct JavaScript (it is ensured that you are actually talking to them via TLS), otherwise you can also host the code locally of course or write your own client.

The crypto is very similar to Bitcoin (same ECDSA curve, but seems like "soon" DJB's ed25519 curve will also be supported for signatures/keys, same base58 encoding primitives for addresses and other user-exposed binary data), so it shouldn't be too hard for you to verify what happens behind the scenes.

Authorities are not holding your keys and (unless you have a very simple user<-->password combo) also can't access them if they get a hold of the blobvault. Even if you own a private key of a third party, you can not reverse transactions (just like in Bitcoin), you can only transfer or trade what you already own, not what you have owned at a certain point in time.

Privacy is a different matter, since Ripple operates on balances, not references to previous transaction outputs it is relatively hard to "launder" money there compared to e.g. CoinJoin. This might chnage in the future, since the next larger feature is going to be Multisign afaik. which would allow several accounts to bundle their payments and thus obscuring individual money flows. In general it is possible to buy XRP anonymously and keep your account off the radar, once you really want to use Ripple itself though and not just play around with XRP, you might have a similarly hard time as with using Bitcoin really anonymous.