Almost definitely it will just work as you've described. Older hardware needed
patch cables to speak with switches and
crossover cables to speak directly to other nics, but that was before nics could detect and automatically switch their input/ouput lines as needed. If you're working with equipment less than say 8 or so years old, you have nothing to fear.
For intranet-only communications you will need new ips, but not from your service provider. The simplest configuration is simply to arbitrarily pick a private subnet and two static ips and start talking. You'll need to set the
scope of the links so the kernel will route packets correctly.
I did this recently for a private connection to my cable modem via USB over which I did not want the rest of my traffic routed:
ip address add \
dev $NIC \
local 192.168.100.100 \
peer 192.168.100.0/24 \
broadcast 192.168.100.255 \
192.168.100.100 became my computer's static ip through which my computer would then route only requests to 192.168.100.? addresses. With only two machines (and probably I didn't either) you won't need the whole /24 subnet range, though you might find it convenient to keep more than just two open. Any online CIDR calculator can help you solve that pretty quickly.
I doubt it was necessary for me to specify .255 broadcast, but I was skimming man pages and plugging it in as I went - in any case, that's almost definitely what yours will be so it won't hurt. The
scope options are important to restrict traffic appropriately, though, so include them.
Oh, I've almost forgot - merely setting the ip isnt all you need - you'll also need to bring up the links after doing so.
ip link set dev $NIC up
You can then (or any time, say, maybe, to find the value of $NIC for instance) check it with :
ip address show
You can have a look at the effect this had on your routing table with: