This would include memories you have also. If you cannot prove it happened you do not mention it to others as if it can?
I have quite a number of interesting memories which I remember, but which I can't demonstrate to be true, so I don't represent them as true to others. Is that really so difficult?
And political beliefs you have. If you cannot prove they are correct - and note this is different from presenting arguments to back it up - you don't share political ideas with others.
You seem to have great difficulty distinguishing between shareing ideas with others and representing those ideas as being true when they can't be shown to be true. There is nothing wrong with sharing political ideas or any other kind with people, as long as I'm not misrepresenting them.
If I have seen a burglar in my house...
And if you just think you saw your neighbor in your house and then spread it all over town and then find out he wasn't even in town that night?
You shouldn't love your opinion more than the truth.
If you do take 'working' as a form of proof
Working is a proof of functionality. If I say I know how to build a bridge and then do so, my claim is verified. But knowing how to build a bridge is related, but not the same as, knowing bridge building. For a long time people in Europe knew how to build a particular bridge, but the didn't know bridge building so any place that didn't fit the bridge they knew, they couldn't span.
how a belief is working for another person
This is a different meaning of the term "working." Also known as "working out" it is a measure of how it fits with their other beliefs and if it is causing them trouble. It is not proof of any particular functionality like a working bridge would be.
it is not or that they 'should' have another belief - also somethign very hard to prove
This is not nearly has hard as people would like it to be for most beliefs. For example only some one grossly ignorant or purposfully stupid fails to accept evolution at this time.
Another example would be the racist "tea baggers."
you jump to the assumption that we can immediately tell whether their belief is true or false. Often we cannot.
Can we immediatley? Sometimes.
Is it imparitive in all cases? no.
Is it possible in all cases? no.
Does that mean we shouldn't make the effect? Don't be silly. Like it or not, the truth matters and religious people have degraded the meaning of truth for far, far, far too long so that it is eroding the public understanding in ways that are becoming dangerous to society and even life in general.
And in many circumstances I have beliefs that I cannot prove to others
And that's fine as long as they are represented as such and you seek to resolve those issue when you can.