If you are signed in to multiple locations (e.g. WLM client, WP7, Windows 8 Messenger) the message will appear on wherever you're signed in to. I assume if a contact is offline, the message will be delivered to the first place he/she signs in to thereafter.
I noticed that WP7 won't receive messages if you're actively chatting on the WLM client, but if you stop chatting for a while and you receive a message, it'll be sent to both places. Which is pretty good so your phone doesn't get bombarded when you're chatting on the desktop, and also for resuming conversations on your phone, even if you left your WLM client on. On the other hand, if you're only chatting on WP7, your WLM client will still show both yours and your contact's messages.
I noticed that right when Mango went live (made a few videos demonstrating it). The WP7 WLM client is very volatile when it comes to Multiple Points of Presense support to the point of being useless when you move between different device form factors a lot. All of Microsoft's other Official Live Messenger clients work properly, though (Windows Mobile, iOS). Only WP7 has this issue, so it's definitely not a normal behavior for an IM client (WLM, Yahoo, Google Talk, iMessage, whatever they all generally work the same except for this issue on WP7). It's actually an issue, but some people are trying to assist off as a "feature." I wonder if it's logged as such at the support site...
For me even my MMORPG game has WLM built into it, so it's one that constantly plagues me, but I get around it by simply not using the WP7 device for WLM, and using the iTouch instead (I just don't sign in on my Windows Phone, since it messes up how conversations end up on the other devices/clients that work properly).
Additionally, The reason why iMessage is superior is because it does the same thing an IM service does, except you don't have to log into it at all - you just have to turn the phone on. WLM requires you to be logged in, even on your device, and IM clients can drop connections at times causing you to have to toggle the online status or sign in again. Clients like BBM and iMessage don't have this issue because of the way the messages are pushed directly to the device.
iDevices and Macs are so prolific these days that you can reliably downgrade your SMS/MMS plan just because of it in many cases.