I have heard from some people that scriptures sanction marriage for girls before puberty. Some people also say that it is sinful for a girl to marry after puberty and that such girls should atone for them. Are these people correct from the point of scriptures? If yes, is there a prayaschitta for this? What is the prayaschitta?
It is not a sin for the girl. It is a sin for the father. Because, as per the Smritis, a young girl is not allowed to act independently or take independent decisions.
At that time, when marriage is supposed to occur, she should be under the control and protection of her father and so it's the father's responsibility to marry her off at the correct time.
Manu Smriti 5.148. In childhood a female must be subject to her father, in youth to her husband, when her lord is dead to her sons; a woman must never be independent.
So, by not marrying her off at the prescribed time, it's the father who incurs guilt. The girl is not held responsible.
Manu Smriti 9.4. Reprehensible is the father who gives not (his daughter in marriage) at the proper time; reprehensible is the husband who approaches not (his wife in due season), and reprehensible is the son who does not protect his mother after her husband has died.
As per the Yama Smriti, apart from the father, the girl's mother and her eldest brother also incur the guilt.
The father who does not, give away [in marriage] his maiden-daughter after she has attained the twelfth year, drinks her menstrual month blood, after month. (22)
By beholding a maiden in [her] menses, her mother, and father, her eldest brother these three go to hell (23)
Yama Smriti 1-22,23
So, as you can see, nothing is being said for the girl. Only the parents and eldest brother are held responsible.
And, if a girl is not married off even after attaining the right age for it, then she can act independently and choose for her a husband.
Manu Smriti 9.91. If, being not given in marriage, she herself seeks a husband, she incurs no guilt, nor (does) he whom she weds.
it is sinful for a girl to marry after puberty
First of all, in accordance with ancient Vedic institutions, girls from respectable families do not "marry" by themselves. They are given through the institution of marriage (kanyaa-daanam) to a suitable bridegroom by elders of the family (specifically the father and mother).
Coming to the point, technically speaking, they (girls) need not perform prayashchitta or atonement. It is the father of the girl as well as the bridegroom that have to perform suitable prayashchittas. Each different kalpa differs in the amount of penance for atonement and that is something that too specific to be stated here.
In general, daanam (donation) of cows (Go-daanam) as atonement for giving one's daughter in marriage after she has reached puberty, is common throughout all schools of the Vedas.
It is however to be noted that this is a very ancient practice that is prevalent in none but very orthodox families only.
Coming to the question; it is not a heinous sin to be married after puberty.