What would you do if you saw someone sitting alone at a lunch table?

Sit down quietly and eat next to them. No conversation unless that person instigated it. We can be alone or together at the same table.

I would respect their privacy.

Probably leave them alone. Honestly, I wish that's what people did for me. I just want to eat without random strangers constantly annoying me ;-;

I look for alone people, since I'm alone. When I started talking if they don't respond I leave them alone and eat my food. There's always a next time and next person.

If I knew them, even a little bit, I would ask them if they wanted to sit with me, otherwise, I'd leave them alone.

It depends on the vibe I get from the person. If they look lonely, I may ask to join them. Most often though, I will find my own empty table and eat by myself.

Shoot spitballs at them from another table.

A stranger .. I would leave alone. I don't know if they prefer it or not. I wouldn't want to make them feel more uncomfortable by putting them on the spot. I would be a stranger approaching them .. And forcing them to answer to me. 

A friend .. I would approach and ask if I could join them

I'd ask if company is needed, if so I'd sit with him/her, the other day he is eating with my whole crew

I would ask if they wanted some company and if they say yes I would stay and chat with them and offer them and opportunity to talk of they needed to. If they said no I would tell them where I will be if they change their mind.

come sit by them.

I would go up to her/him and say "Hey, can I sit here. It looks like you're sitting alone, I don't like seeing people sitting alone. I can be your friend, if you want?" I would talk to them, tell them about myself and tell them funny jokes. Because I love making people laugh! It won't matter how they look. I won't care what other people would think of me for being their friend and talking to them.

I usually assume that they are probably like me---and want to eat, relax, and think without interference or distraction.

First i will offer her to join us and if she don't want to join us then it's okay i will respect her privacy

Haha! That's me right there! ... But, if I weren't, I would invite the person to join me/my group. If they decline and seem shy or nervous about it, I don't care. I'm gonna plop myself down on the table and sit with them the whole way through and try to start a conversation. If they seem to have no interest, then I guess I'll just make an excuse to leave the person alone.

It really depends on what your agenda is. If you want to make contact you'll have to analyse his/her body signals to determine if he/she's even open to company and if not prepare yourself for the expected reaction.  You should check out trump techniques, really work