The issue you are going to face is similar to any database-based solution: the issue is ownership over a file and if two systems are trying to change a file at the same time. This can often corrupt a database for systems that are not designed with this concurrency in mind, and most systems are not designed with this in mind. As mentioned above, there are some higher end solutions that are designed with this in mind, but they are expensive and typically designed with an agency in mind.

Assuming you do not wish to purchase an agency solution, Lightroom would do fine, if you did not wish to share the database and each others edits. In this case, you would simply share an external drive with the photos themselves, but each keep your own catalog of edits local to your system. I am not sure if Aperture works this way or not, others can perhaps chime in.

The only alternative to get something close to this function with Lightroom, is for you to share the hard drive of photos as above, but then, use the export/import catalog function of Lightroom to "share' the catalog between machines. So, have your wife export a catalog of images she has edited, and then you import them into your catalog...and so on. Not especially simple, but it will work, as long as you can put up with it.

Of course, if you can get by without sharing edits, then simply put all your images on a network accessible drive, point each copy of Lightroom at this shared drive and edit away. One note: when your wife adds images, they will not automatically be added to your catalog, but you can easily solve this by right clicking on the shared drive link in the Library view, and select 'Synchronize Folder" which will add any new images to your local catalog.