What's the best way to set up a live webshow with multiple Skype video inputs?
The popular VLC media player offers low latencey multicast video streaming solutions. This is not as one-clickish as Skype, but hey, thats what you get for becoming more professional.

Video streamed by VLC can have any properties regarding video resolution and quality - within the limits of your network bandwith of course.

Ususally your PC will be behind a home router which acts as so called NAT-Gateway and firewall. Much to the annoyance of network administrators Skype sees to it automatically that this firewall is cranked open, and multidirectional video casts can flow through. It also handles IP address and port negotiation. VLC does not, as the concept behind it is different (its main use is streaming, not video conferencing).

So for this post I will assume that you are willing to invest a degree of brain juice in creating a clean media flow over the internet. After all you are working on a global technological scale. Do not worry, we still will not need to get into the advanced internet routing stuff.

With VLC you will have to configure your home router manually to present a service running on your computer to the open internet. This is called port forwarding, look it up.

I will not get into details about setting up VLC streaming, as it is pretty much straight forward and you can handle this yourself: Open VLC, press Ctrl S, follow instructions. Best use RTP or HTTP streaming as they are well standardised.
When setting up your VLC to stream media from a webcam, you will configure a network port, on which the webcam stream is offered to the internet. Make sure this port is forwarded by your home router.

Decrease media caching to improve latency, increase caching to prevent stuttering. 200ms will be fine. A h264 stream with 720p resolution will look OK at about 2000 kbps but only if you let VLC handle compression. If you let your webcam handle the compression internally you will need much more bandwith, as the hardware encoders are somewhat limited.

Once both parties (you and your cohost) have set up this stream, you can connect to each other using a media player like VLC, mplayer, Windows Media Player, Quick Time... streaming really is a standard ;-) Any guests you are hosting can connect to this stream too, and so can spectators. VLC also lets you record remote video streams to disk for post production.

To get guests on your screen, you can let them use a tool of their convenience like Skype or Hangouts. Once VLC is using your webcam, you will not be able to offer video from the same cam via Skype. But this will not be necessary, since guests can see you over the VLC stream. You would only need Skype to see them, while video streamed from you and your partner remains fluent.

To have people connect to your stream, they will need to know your public IP address, which you can look up on http://www.whatismyip.com or similar sites. You can use a dyndns service like http://no-ip.biz (or other) to use a convenient DNS name instead of an IP address.

With this description you should be able to get started, ask me anything if you need to know more.