We tried this a few years ago with (IIRC) a feature Yahoo was offering. The SMS would go one way, but not the other. I can't remember which way it worked for, but perhaps things have improved by now. It was a javascript box, so a shared computer should be no problem provided there is no firewall block.
I would suggest that the person with the phone gets something like AIM, YIM, or MSN messenger for their phone. Most of these have ways of sending messages without actually signing into a program. For instance, AIM has a web version (if they still support it). YIM also has a web version, but it also allows you to send a message by clicking on someone's Yahoo profile.
The downside to this would be that the person with the phone would have to be signed into the app unless the program allows offline messaging.
We used Virgin Mobile before I moved. You can get a cheap $10 phone, put minutes on it, activate international calling, and send texts/call. It's not cheap, but I was able to text to the UK (not vice versa) this way. When I had my Virgin phone, I think an international text was $0.25, and it was something like that a minute to call a UK landline. Good for emergencies, and an option, especially if you can afford the top up pay monthly plans.