
austeilen
to hand out
To distribute something to a group of people so that everyone gets a share.
Der Lehrer teilte die Arbeitsblätter an die Schüler aus.
The teacher handed out the worksheets to the students.
to hand out / dish out
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to hand out
To distribute something to a group of people so that everyone gets a share.
Der Lehrer teilte die Arbeitsblätter an die Schüler aus.
The teacher handed out the worksheets to the students.

to dish out
To direct or deliver criticism, insults, or even blows at others.
Im Streit konnte er gut austeilen, aber schlecht einstecken.
In an argument, he was good at dishing it out but bad at taking it.
'Austeilen' is a separable verb. The prefix 'aus-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Die Freiwilligen teilen das Essen aus.
The volunteers hand out the food.
The thing being distributed is in the accusative. The people who receive something are often connected with the preposition 'an' + accusative.
Die Lehrerin teilt die Zeugnisse an die Schüler aus.
The teacher hands out the report cards to the students.
In the literal meaning, 'austeilen' is often combined with things that are distributed to a group. Typical examples are: 'Getränke austeilen' (to hand out drinks), 'Karten austeilen' (to hand out cards), or 'Werbung austeilen' (to hand out advertising flyers).
Vor dem Spiel teilte der Trainer die Trikots aus.
Before the game, the coach handed out the jerseys.
In the figurative meaning, the expression 'austeilen, aber nicht einstecken können' (to dish it out but not be able to take it) is very common. It describes a person who likes to criticize others but cannot tolerate criticism themselves.
Mein Kollege kritisiert ständig alle, aber wehe, man sagt etwas zu ihm. Er kann gut austeilen, aber nicht einstecken.
My colleague constantly criticizes everyone, but just try saying something to him. He can dish it out, but he can't take it.
to distribute / spread
Verteilen and austeilen can both refer to giving something to several people. Austeilen usually emphasizes handing something directly to each person, so that everyone receives something; verteilen is broader and can also mean generally assigning something to people or groups.

Die Lehrerin teilt jedem Schüler ein Arbeitsblatt aus.
The teacher hands out a worksheet to each student.
Die Lehrerin verteilt die Arbeitsblätter an die einzelnen Gruppen.
The teacher distributes the worksheets to the individual groups.