A group of people were being very aloud. The other person arround that partucular group was also being equally loud.
Could you please explain the grammatical rule of "being" here?
If I Just write "A group of people were very aloud." Is it going to change the meaning of the sentence?
Let me start by offering a 'better' sentence but retaining the meaning.
A group of people were making a lot of noise. Another person, near the group, was equally noisy.
'Being very loud' where 'being' is indicating what the group 'were doing' is an adjective; I am being pedantic; you are being curious. 'Being' is the explanatory adjective - without that in each of those cases the meaning would infer a fixed state, whereas with it the meaning infers 'now', not necessarily at any other time.
Note the correct noun is loud - aloud is an adverb. So your last sentence would be correct using 'loud'.

Now the second sentence, corrected: the other person (around is wrong; one person cannot stretch around a group of people - impossible - but a group of people could be around one person, in other words encircling him/her) near to, close to, at the side of, alongside that particular group was equally loud, was also loud, was being equally loud. In the sentence 'also' and 'equally' are really unnecessary; one or the other will suffice.