The Dogecoin client is based off the Litecoin client. The Litecoin client is based off the official Bitcoin client.
Odds are, if a remotely exploitable bug is present in Dogecoin's network service, it will probably be present in Bitcoin's too, and every altcoin that is a direct fork of Bitcoin.
Cryptocurrencies require you to establish P2P connections with other people on the Internet, so if you want the network to thrive you do have to run it on a public network. There is always going to be a risk with any service you run, but Bitcoin's code has been audited by security professionals on quite a few occasions, so the chances of a critical bug that can be remotely exploited against someone simply running the client are fairly low. There's probably a higher chance of there being a critical bug in uTorrent.
Hmmmm, you may be right...i'm not super knowledgeable on this type of thing though. I'll update the picture, but can you give me a explanation I can use?
Basically, the Dogecoin wallet is going to attempt to punch a hole in your firewall with UPnP and any other NAT-busting techniques the author (of Bitcoin) has contrived to enable you to help new nodes to come to sync by sending them blocks from the winning blockchain.
The daemon mode (which listens on an RPC/HTTP port for commands like "send money here") is not going to attempt to run unless you enable a password in your .doge/config (just like bitcoind). RPC clients will need this password to be able to issue commands. If you have one node, then both the client and server will have the same .doge/config and it's secure, assuming they don't guess your password. If you have multiple nodes, this is the way to have a lightweight client that sends commands to your "heavyweight" wallet. If the client doesn't have your RPC password, the client can't do anything that any 'nobody' new node on the network should be able to do.
You can run Dogecoin on public networks, or the Dogecoin wallet has a bug that you should report. Whether it's wise to be in a position to be the first to discover such a bug (by losing all the coins in your wallet), that's another matter, but if there are such bugs in the wallet then it's really not viable, and you should therefore probably sell all your doges immediately.
(If you haven't ever looked at your .doge/config or attempted to run in daemon mode, you won't have an RPC password and your client is also secure, since it won't accept commands that are not authenticated.)
I don't think you should be allowing the Doge Client to run in public networks. Am I wrong?