Can the an American iPhone 5S work in Europe and vice-versa?
It's not going to work because Sprint will not let you add her phone to their network. Sprint limits subscribers to phones specifically sold for the Sprint network. Verizon has a similar policy.

You may, however, be able to get the phone to work with a different carrier. Many European companies typically sell unlocked phones, so if the phone was purchased in Europe, it may already be unlocked. In that case, the only issue would be to find a network that supports the phone in the United States. Considering that European countries often use GSM networks, you may be able to use AT&amp;T, T-Mobile, and possibly some other carriers. If you need to use the phone for only a short time in the United Staes, you may want to buying an AT&amp;T GoPhone sim card or a similar contractless service from another carrier.

If you do, however, need to use the Sprint network then you should keep in mind that the two phones support different LTE bands so it's likely that you both would lose access to LTE. It's not certain because there is some overlap but it's quite likely.

https://www.apple.com/iphone/LTE/

Here's the FAQ for Sprint unlocking:

http://www.sprint.com/legal/unlocking_policy.html