How to build research collaboration? I am an assistant professor of chemistry. My department strongly values international collaborations. I contacted some research groups working on the same topics as mine, but half of them wanted my fund, and the other half thought I want theirs.

My imagination of research collaboration was that two groups do parts of a research idea by their available funds, and then preparing a joint paper by combining the results.

For example, a common suggestion I received is that they can host a PhD student or a postdoc fellow (for a minimum period of one year for the sake of official paperwork and adaptability) if I have a fund. How can I fund someone to work somewhere else on a project different from my funded project? And how is it a collaboration after all?

In my experience, it is very difficult to spend research funds in another country. It is even impractical to send joint proposals (unless there are governmental agreements between two countries). I wonder how do researchers from various countries plan international collaborations (which is apparently very common based on the joint papers)?
Generally, and as many questions here on Academia.SE reflect, collaboration typically does not grow from blind-mailing people that you have never spoken to and which, presumably, have little to no idea who you actually are. The fact that many of them thought you are interested in leeching of their funds is not surprising, because that's what a lot of these blind mails boil down to.

Dawn is on the right track in a comment - before you can have collaborations, you need connections. A common way to establish connections is by going to conferences and chatting with people - initially not even necessarily about common work, although the discussion often tends to go there fairly naturally.

Optimally, you would have started this process long before you were an Assistant Professor - in fact, arguably one of the key tasks of an academic supervisor lies in introducing their students into their academic network, exactly so that they have an easier time bootstrapping future collaborations of their own.

Even if you have no real network right now, you can probably still fix this, but you should see this as a marathon, not a sprint. You will need to go to a number of conferences, talk to a lot of people, and and generally be present in the community for some time before your name starts to come up naturally when people think about who to collaborate with on a given topic.

However, once you have started to be somewhat known in the community, even as fast as after your first conference, you can still approach the people that you talked to there with concrete ideas for collaboration. Be prepared that especially initially you may need to put in a bit more effort than they do, particularly if you collaborate with somebody more senior than you. This does not mean that you should do all the work and just put them on the paper, but if a collaboration is your idea it is often assumed that you are also driving the project and ask your collaborator(s) for concrete, actionable input.