game of thrones - How do Southern people discern Night's Watch deserters from truthful brothers sent south?

It has already been answered that it is possible that Members on business of the Watch would carry identification and authorization documents with them.

Few things have not been discussed yet:

Size of the North

North was almost as big as rest of the seven Kingdoms combined.

From Eddard's POV in AGOT:

“I was starting to think we would never reach Winterfell,” Robert
complained as they descended. “In the south, the way they talk about
my Seven Kingdoms, a man forgets that your part is as big as the other
six combined.”

Population is very sparse in the north. From Eddard's POV in AGOT:

Robert snorted. “Bogs and forests and fields, and scarcely a decent
inn north of the Neck. I’ve never seen such a vast emptiness. Where
are all your people?”

Presumably, people live in closely knitted communities, mostly in villages because there is only one true city in North which is White Harbor.

In villages, people tend to know who is who. So if a stranger walked in on some settlement, People will know him for an outsider and Questions will be asked. Questions will lead to suspicion and thus eventual capture of the deserter before the deserter ever made it out of the North.

Necessity to steal provisions

A person wearing black clothes south of the neck was always viewed with suspicion and people kept an eye on their movements. That was because NW men seldom had business down there except the few wandering recruiters like Yoren. From Jon's POV in AGOT:

A stranger wearing black was viewed with cold suspicion in every
village and holdfast north of the Neck, and men would soon be watching
for him.

That makes it paramount for the deserter to steal clothes which may lead to hue and cry by the owner of the stolen clothes, leading to capture of the deserter. This was not limited to clothing only. If by mischance, the deserter had any bit of black equipment on him, he would be done for. From same chapter in AGOT:

His longsword and dagger were sheathed in black moleskin, and the
hauberk and coif in his saddlebag were black ringmail. Any bit of it
could mean his death if he were taken.

Also, The deserter would have to find food as well. That would involve stealing from villages or hunting in the woods. Hunting had the risk of running into some Lord's hunting party and stealing had the risk of the owner raising an alarm over missing food, animals or goods.

Warrants issued from Castle Black

As soon as Castle Black became aware of the desertion, warrants would be issued for the arrest of the deserters which may include description of the man's physical appearance and other distinctive traits. The warrants would be received by all the concerned lords who would then forward them to their patrols and subjects. Those warrants were sent by ravens so the word spread faster than the deserter could move. From Jon's POV in AGOT:

Once Maester Aemon’s ravens took flight, Jon knew he would find no
safe haven.

The necessity to remain out of sight

Even if someone made it all the way to South without encountering any problems, he would have to live a lie and change his entire life's story. But as soon as he caught attention of someone, it was unlikely that the story would hold. From Jon's POV in AGOT:

He was who he was; Jon Snow, bastard and oathbreaker, motherless,
friendless, and damned. For the rest of his life-however long that
might be-he would be condemned to be an outsider, the silent man
standing in the shadows who dares not speak his true name. Wherever he
might go throughout the Seven Kingdoms, he would need to live a lie,
lest every man’s hand be raised against him.

Conclusion

1. Northerns mostly live in small communities and thus are bound to know all the residents of their community. They would instantly know an outsider. North is very huge which means that a deserter will have to go through a lot of communities and answer a lot of Questions which can lead to suspicion by people.


2. To survive, a deserter will have to steal or rob. That would put undesired attention on him and might result in arrest by some local Lord.


3. Castle Black issues warrants to all concerned lords in case someone deserts. So most likely he would be taken by patrols on his way south.


4. South of the neck, every man in black was viewed with suspicion because Night's watch usually had no business there except for recruiting or gaining grants which was done by wandering crows. So if there is a man in black without any document authorizing him to ask for prisoners as new recruits or for begging aid, It is probable that he is a deserter.