Is it a valid concern that a PhD student runs out of research ideas?

Research ideas tend to grow exponentially.

• At the beginning, Alex is a student who doesn't know what's already been done -- no ideas (though maybe some interests)
• As an undergraduate, Alex will start to have ideas based on what he reads and his experiences. Many of these will be off-topic, already done, or impossible.
• Then Alex will spend time doing research "at the cutting edge" -- usually on someone else ideas (i.e., his advisor's). He'll now start to understand where the gaps are and what ideas could be developed, adapted, or applied to plug these gaps. Some of these ideas could be valuable.
• As Alex continues to read the journals and perform his own research, he'll identify more and more ideas. The limiting factor is time: having ideas may be fast, but taking an idea from conception to publication always takes a long time, so Alex normally has a long list of ideas in the back of his mind.

So yes, Alex will gain by discussing his ideas with his advisor. In most cases though, students start by work on their advisor's ideas, which they gradually make their own and begin to supplement with their own ideas as they mature.

I am now at my 5th year as post-doc, and I do have much more ideas than time now. I share them with PhD students and coworkers, and try to make joint papers, as it is beneficial for both of us.

Ideas are cheap -- usually free. (Or even cheaper than free, having negative values - in many areas creators of ideas have to pay people to attend to them, like paying agents to plug their music and books.) Most people already have far too many of their own ideas that they don't have time to do anything with. It's unlikely they would get any benefit from stealing any more from anyone else. Actually doing stuff is hard and involves money, time, effort and reward.