I tell the person that I appreciate what they did, might add a note about why I appreciate it if I think they need the affirmation.
Sometimes, depending on the situation and what it is I'm appreciating them
for - if it's just something they did for me that I appreciate - then I'm likely to get them a gift or render some service in return that makes us even. Keep the slate clean, so to speak until I know if they're a slate-watcher
But if we're talking about appreciating someone on a more personal level rather than just stuff they do for me, basically, I invite them to join me in my leisure time.
It has been noticed in the past that simply not turning someone away when they come to me is not necessarily a great sign that I appreciate their company or even like them at all - it could very well simply be my sense of duty to humanity in general that makes me unwilling to turn away a person in any kind of need even if it's my least favourite person. But if I actually take the intiative myself and invite them, then that is a really big deal, that's me saying: I officially want you to be my friend because I totally appreciate the rockin' person you are. You have been headhunted!

I hope that people would notice the significance of this and not keep fishing around for verbal compliments and expressions of appreciation... I will toss them out if I sense a person needs one, but it won't take many times of having to do that before I start to go cold on them.
I wouldn't invite someone I didn't truly appreciate just to avoid being bored and alone, cos I know that whether I invite people or not, there's an above even chance that someone or other will turn up any damned way.
