Is it possible to tell:
They have begun (to) being checked.
The meaning: The police started to check people some time ago. But I can not mention "police" in the sentence.
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In the first place, begin takes two sorts of complement, which you may have confused:
Be careful not to combine the two forms: begin to checking is impossible!
In the second place, They have begun does not designate a "past" event: it is a present tense, the present perfect, which designates a current state which originated in a past event.
The meaning of a perfect construction (and consequently the propriety of using it) depends on the context, including the verbs involved. This gets very tricky with begin, which has two different senses:
Without more context it's not clear which meaning you intend here. I suspect you mean the first: checking started some time ago (and presumably continues today). If that's the case, begin designates an event which happened entirely in the past; it is only the checking which continues. Accordingly you should use the simple past:
Police began [checking/to check] people, or in the passive
People began [being/to be] checked.
But it is also possible that you mean the second: the police started their checking activity in the past but are still in process of extending that to all the people involved. In that case you would properly use the present perfect:
Police have begun [checking/to check] people, or in the passive
People have begun [being/to be] checked.
Note, by the way, that these examples do not exhaust all the contexts in which the contrast between simple past and present perfect arises. This is a very complicated matter.
"They have begun (to) being checked" is incorrect.
If "they" refers to the subject being checked by the police, you could say:
They have begun to be checked.
If "they" actually refers to the police carrying out the check, you could say:
They have begun to check.
Both these statements assume that you have previously established who "they" are, who or what is being checked, and what that check entails.
"They have begun to be checked [by the police]" is a well-formed past passive sentence (with or without "by the police").
"They have begun being checked" is probably fine as well.
"They were being checked [by the police]" is also fine.
Edit: I discussed emending your text in a way that doesn't work, but there is no need for that discussion, so I have removed it.