2 Apple IDs on one device?
Solution 1:
Yes, you can have apps installed from different Apple Id's. Simply log in with whatever account you both want to use for purchasing. You can log out and back in with a different App Store account at any time.
I have multiple accounts, and multiple devices sharing applications.
To follow up the other responses, your App Store account can be separate from your iCloud, Messages, and FaceTime accounts. I use a separate account (Apple ID) for iTunes purchases than my iCloud account.
Some people even split things the opposite way - sharing email, FaceTime and messages but having different store accounts. It's really up to whatever works best for you and your devices.
Solution 2:
This may help. It's from the Apple iTunes agreement we all have to 'read' and sign...
Association of Associated Devices is subject to the following terms:
(i) You may auto-download iTunes Auto-Delivery Content or download previously-purchased iTunes Eligible Content from an Account on up to 10 Associated Devices, provided no more than 5 are iTunes-authorized computers.
(ii) An Associated Device can be associated with only one Account at any given time.
(iii) You may switch an Associated Device to a different Account only once every 90 days.
(iv) You may download previously-purchased free content onto an unlimited number of devices while it is free on the iTunes Service, but on no more than 5 iTunes-authorized computers.
An Apple TV is not an “Associated Device.” However, TV show iTunes Products and purchased (i.e. not rented) movies iTunes Products may be played back on compatible Apple TVs, provided that you may only play back any such TV show or movie on a limited number of Apple TVs at the same time.
Hope this helps.
Solution 3:
Recently Apple introduced Family Sharing.
This means you can share all your purchased apps, books, music, etc with up to 6 people you define yourself to be family. Apps or other media you purchase are not automatically downloaded on their devices, but they do get free access to it. On the side you can share photo albums and calendars, but that goes beyond your question. Just have a look for yourself what it can all do: https://www.apple.com/ios/whats-new/family-sharing/
Solution 4:
I'm not entirely sure I follow your reasoning: having "two Apple IDs logged in at once" doesn't solve the problem of purchases occurring in the context of an Apple ID - there is no way to purchase an app with two different IDs without both accounts being charged. (Please elaborate if I'm misunderstanding what you're asking :) )
However, across multiple devices, you can use a single Apple ID for purchasing, and separate Apple IDs for iCloud, if you want one source of purchases but separate sources of data / backups on each device.
The settings are distinct: In iOS, you pick the Apple ID used with the iTunes Store via Settings > Store and for iCloud via Settings > iCloud.
So, for example, you could both use your account for the Store, and have the purchases made on each of your devices.
That said, there is no easy way to use separate accounts with iCloud on one device, as setting the device to use Account #2 will blow away Account #1's data, making it difficult to truly share a single iOS device unless you don't care about separating your contacts, Photo Stream, etc.
Solution 5:
Yes, you can actually have 7 different Apple IDs on one iDevice. See this question I asked previously:
Can I have different Apple IDs on my iPhone 4S?
And yes, you use the same Apple ID on multiple iDevices. This would allow you to share apps.