Understanding a "streamer" setup in regards to graphics card setup

Hi everyone,
To make a short story shorter, my girlfriend's brother passed away recently, so I acquired his setup.
This consists of an 8th gen i7, 32gb ddr4, gtx 1080ti sc, and an Elgato HD60 Pro capture card.

I'll admit I had no idea what the elgato was when I first spotted it, and until recently I knew literally nothing about streaming, and streaming setups.

Mind you, the capture card is installed in the same "gaming PC". I did some research and noticed that it's common for streamers to have their gaming pc, and then a secondary "streaming PC". Is this necessary?

I have another PC that's currently my linux server that I could potentially re-purpose to be used as a streaming PC with the elgato installed.

I did play around with it, and I found it somewhat confusing. Obviously it's got an input and an output port... so I connected the output from my GPU, to the input on the capture, then the output on the capture card to my monitor.
Is this not a common setup? It seemed to work ok like that...

Would I benefit from setting up a secondary PC with the elgato installed?
What are the pro's and cons of running the capture card from a secondary pc, vs. installed directly in the gaming rig?

As you may notice I'm pretty much a noob when it comes to the practice of streaming, but it seems like it might be a fun hobby and I like to tinker, so I'm wondering what's the proper way to go about setting this up?

Thank you all for any input!

I should have added that I currently use an HDMI switch and a single monitor to switch between each of my computer setups.

I would therefore have one hdmi cable coming from the Elgato output into the hdmi switch to handle the "gaming pc" video output, and then I would have another hdmi cable coming from the "steaming PC's" gpu to handle video output from that machine.
Then I could continue to switch between machines like I'm currently doing, correct?

It sounds like you have it set up correctly.

The advantage, generally, to using a second PC for handling the stream is that it frees the gaming PC from the burden of doing the game and processing stream video at the same time. With the many-core CPUs that exist now, this is less of an issue than it was three or four years ago, but there's still something to be said for it if you're anal retentive about getting every last frame out of your hardware.

More practically, if you're doing the sort of streaming where you interact with the audience, I would think the more useful advantage of having a second PC would be that it would also have its own monitor, so you can keep tabs on chat and so forth without the need to alt-tab out of the game.

It really depends on how serious you are with streaming.

Actually you don't even need a capture card for streaming, the GTX1080Ti is more than capable by itself. My kid has done some streaming only with a GTX1070Ti

But if you want to do PiP and interactions, green screen and stuff, then a 2nd PC is recommended.

If the guy was trying for max quality in 1 player games or max FPS in shooters it’s be common to use a streaming pc.
I didn’t like the impact recording and streaming dealt me, even if I could choose 1 workflow to use nvenc, regardless of content quality.
Ive gotten into my buddy’s 3950x/2080ti build that he has sunk tons of time one tuning.
You can still feel streaming and recording drag on his rig so he has gone back to 2 pc setup.

Elgato card is a cleaner way to send the game to the 2nd pc, especially if it gets hammered cpu encoding.

OBS can send the stream to a second computer on the same network which can perform recording/transcoding/streaming and not impact your primary gaming PCs performance in a noticeable way. It is not a bad path for someone to take if they plan to build a new PC or pickup another one for cheap.

However, other hardware exist that is cheaper (sometimes) and can do that job. Maybe the software/dual system setup would allow you to power through an codec change while other hardware may not.

If he had the card installed in his gaming PC, it's possible that he used it to stream/capture something else such as consoles and then did single PC streaming for the rest of his stuff. I'd personally use it in a second PC though like you plan to do.