I have installed two Ubuntu partitions and they appear to be using two different swap partitions. Is this a safety thing or is it a waste of space? If the latter, how do I set up one Ubuntu to use the other's swap partition?
It's a waste of space - when installing the second Ubuntu you could've just pointed it to the existing swap partition. Actually, I even think the installer automatically configures an existing swap partition to be used if it can find one.
To fix the situation (well, if you consider it a problem), you need to open /etc/fstab file of one of the ubuntus, locate the line which mentions "swap" and change the device name (or partition UUID) there.
The line would look something like
# /dev/sda2 UUID=69222dd6-31ea-488e-a4c1-8f5a5fbba098 none swap sw 0 0
so you can change it to
/dev/sdb5 none swap sw 0 0
or, even better, find the UUID of the swap partition and use it instead:
sudo blkid /dev/sdb5 /dev/sdb5: LABEL="SWAP-sdb5" UUID="69222dd6-31ea-488e-a4c1-8f5a5faac456" TYPE="swap"