J.W.
M.,
If that had been my daughter your son had hit and pushed I can assure you I would have said something to him. You should have addressed the situation right then and there. Who cares who is around. You need to teach him how to express himself with words not hitting. Part of his problem, is his age but an almost 3 year old should have some verbal communication skills. He hits in part because he can't fully express what he wants. He also seems to do it just because he can.
Every child goes through a hitting stage, some more than others, and there was a time when my daughter was around your sons age that she would hit me when I put her in the car seat. And she can hit hard. But when it happened she got punished. Sometimes she made me so mad I couldn't say anything. Sometimes the silent treatment is worse than an actual punishment. Staying calm in a situation will help keep it from escalating but a problem needs to be address when it happens.
Talk to your childs doctor, take a parenting class, read parenting magazines, look up info on the internet about how to handle tantrums. All can help you deal with the situation better.