Am I able to share contact information on the desktop version?

Absolutely! 

Within any event, you can see the guest list on the right side. If you tap the three dots beside someone's name, you'll see an option to "Connect". Once you click, a box will pop up prompting you to choose what you'd like to share with that specific person. Select the pieces of contact info you'd like, hit "Share", and your connection request will be fired right off! 

Similarly, if you are not in an event, but are viewing someone's profile, you can click "Connect" under his/her name and photo. The same prompt box will appear and your connection request will be sent in no time!