Scout and Jem were generally allowed to roam around their neighborhood and around Maycomb.  Some of this had to do with the way life was in small towns in the 1930s.  Some of it also had to do with the fact that Atticus Finch trusted his children to have some level of independence.

There were several examples of Jem and Scout's independence in the book.  The siblings played throughout the neighborhood and spent a considerable amount of time lurking around the Radley house with little notice.  They snuck out of their house to go check on Atticus when he sat up all night outside the jail.  When Atticus was defending Tom Robinson at the trial, Scout and Jem snuck inside the courthouse to observe.  They did this even though Jem knew that "Atticus might not like it if he [saw them]" at the courthouse."  

What are some examples of how characters like Atticus, Scout and Boo show independence in To Kill a Mockingbird?

CHARACTER INDEPENDENCE IN TO KILL A MOCKINGBIRD

Atticus.  Atticus is occasionally forced to compromise with several of the characters in the novel (particularly Scout and Alexandra), but he puts his foot down when his sister tries to force him to fire Calpurnia. He tells her that "you'll simply have to accept things the way they are."  (Chapter 14)  He defies the wishes of most of the people of Maycomb by agreeing to defend a black man accused of raping a white woman; and when Tom's life is threatened by the appearance of the lynch mob, Atticus risks his own life by standing up to the men alone  (Chapter 15).

Scout.  Scout is most often a follower, tagging along after Jem and Dill during their raids on the Radley property, but she shows an independent streak when it comes to decision-making: She recognizes the academic and moral deficiencies of her teachers; she realizes that the gossip provided by Miss Stephanie is mostly untrue; and she sees that the supposedly "devout" ladies of the missionary circle are both hypocritical and unladylike.

Boo Radley.  The reclusive Boo makes the bold decision to finally expose himself when he sees that "his children" are threatened by Bob Ewell on Halloween  (Chapter 28). Previously, he risked public exposure when he placed a blanket upon Scout's shoulders to protect her from the cold on the night of Miss Maudie's house fire  (Chapter 8).

Tom Robinson.  Tom knows better than to enter a white man's house, but his inherent kindness and the sympathy he feels for Mayella Ewell causes him to risk his own safety by performing an act of kindness. His final act is also one of independence, deciding that he was "tired of white man's chances and preferred to take his own" when he tried to escape  (Chapter 24).