I'm assuming that both of your iPhones are logged in with the same Apple Account. Is that correct? If so, it doesn't matter which phone you used to pair the tag with.
Once both phones are outside of the Bluetooth range of this tag, the tag will rely on the Apple Find My network to send it ID to. In turn, any iPhone that get's within range will pick up that ID, and relay it, along with that phone's GPS location, to the Apple servers over the Internet.
What I'm not 100% sure about is this particular situation where you leave one of your phones nearby the tag.
If I had to venture a guess, that phone would still receive the tag's ID by BLE or Bluetooth, but unless that phone has an active Internet connection, it won't be able to relay it to those Apple servers ... thus, you would not see any updates.
I suggest you hang in there and see if someone else chimes in with a better answer for you.