Site-Only Methods
Unfortunately, as you have discovered, there's no easy way to do this, not if you're just wanting to quickly scan through a bunch of cards.
The easiest way to do it for a single random card is to click that card's link (opens its detail page in a new tab), then type Ctrl+W to close the tab and go right back to your friend's inventory. (I love keyboard shortcuts! They make things like that quick.)
If instead you're browsing through multiple cards that are in an easily-definable subset (like a specific edition, or foils), I'd probably find it easiest to open my and my friend's inventory in separate tabs, filter each to show just the desired subset (edition, foils, whatever), apply the same sort, then yes.... flip back and forth between the tabs. :\
That being said.... What's your specific use case? I'm assuming your goal is something that Deckbox's trade functionality doesn't fulfill....
Possible Tool
The reason I ask is that I've created Excel spreadsheets for almost every set up through Khans block (and it would be very simple to create sheets for later sets/blocks). These sheets allow you to see how many of each card in that set is listed in multiple inventories. It works by typing in the filename of an open inventory file, which then populates that column with the counts from that inventory. You can then "lock" that column, so that the values won't disappear when you close the inventory file. You could compare as many inventories this way as you want.
If you think you'd benefit from these sheets, let me know, and I'll put them all in a ZIP and make them available for download. The obvious limitation is that you DO have to go set by set. That's why I asked how you're trying to compare inventories. If you're already doing it this way, then this should be perfect for you!
Background
I created these spreadsheets because I had originally managed multiple physical inventories, and had a separate digital inventory for each of them, but I ultimately combined the physical inventories. So in order to spot check myself (since I did a lot of the work when I was sleep-deprived!), I created these sheets so that I could go through an edition of already-combined physical cards, and compare the quantities to the Total column in my sheets. Yes, I'm a bit OCD about it.
EDIT: I should note that I still keep foils and foreign cards in separate inventories, both physically and electronically, so if you have e.g. 1 regular, 1 foil, and 1 foreign version of a card, these sheets are going to show that you have 3 of that card. I could potentially customize this functionality depending on what you were needing.
EDIT 2: I just took the time to glance at your inventory, and I noticed that all of your cards are from Shadows Over Innistrad. So tomorrow, I'll create an SOI version and upload it, then you can get more of an idea what I mean!
Last edited by meldon44 (2017-08-24 08:05:11)