Yes, you can, but there are restrictions in both cases.
Apple: You are allowed to designate up to 100 devices that can run your app without it being in the app store. You have to collect the UDID’s for each device you want the app to run on, put them in your Apple Provisioning Portal, then create a Provisioning Profile called an “AdHoc” profile that includes the devices you want it to run on. Next, build your app using that profile and then you have to build an .ipa file (beyond the scope of this post). Then you can give the .ipa file to those devices and they can use iTunes to install the app on their device.
Android: Android devices can install any .apk file from anywhere they can find it. It can be from a web page, an email, drop box, etc. The catch is typically the device owner has to either be in “Developer Mode” or “Enable App Installs from non approved sources”. Getting your people to get their devices into a mode where you can “side load” apps is the trick.