In English there's expressions like [to hear sb. saying sth.] or [to see sb. doing sth.], but when it comes to German, I found it difficult to form the sentences short and natural. Please also tell me if there's a difference between formal and casual expressions. Thank you!
1. to hear sb. saying sth.
This can basically be translated as
jdn. etw. sagen hören
Ich hörte sie sagen, dass sie Hilfe benötigt. – I heard her saying that she needs help.
Ich hörte ihn seinen berühmten Satz sagen. – I heard him saying his famous sentence.
Sie war sehr überrascht, mich das sagen zu hören, weil ... – She was very surprised to hear me saying so because ...
2. to see sb. doing sth.
That can basically be translated as
jdn. etw. tun sehen
This may sound a little bit odd, but it becomes clearer with some examples:
Ich sah ihn essen. – I saw him eating.
Ich sah sie tanzen. – I saw her dancing.
Unsere Familie und Freunde sind froh, uns etwas Hilfreiches für die Gesellschaft tun zu sehen. – Our family and friends are glad to see us doing something helpful for the society.
Difference between Formal and Casual Expressions
You can use these phrases for both, formal and casual environments. However, in the latter people tend to use "dass"-clauses for more complex sentences, for example:
Ich hörte, dass er seinen berühmten Satz sagte. – I heard him saying his famous sentence.
Ich sah, dass er ihr half. – I saw him helping her.
Unsere Familie und Freunde sind froh zu sehen, dass wir etwas Hilfreiches für die Gesellschaft tun. – Our family and friends are glad to see us doing something helpful for the society.
"dass"-clauses may also be used in a formal environment, however they do not sound quite as elegant in written language in my opinion.
As an addition to the answer by tavkomann, I'd like to add the more colloquial expression
Ich habe gesehen / gehört, wie...
This is used more often than "Ich hörte / sah, dass..." It is, however, limited to the literal meaning of the verbs see and hear. For instance:
Ich habe gesehen, wie er fünf Cheeseburger direkt hintereinander gegessen hat.
Ich habe gehört, wie der Lkw direkt unter dem Fenster vorbeigerumpelt ist.
Exactly as on English: I habe ihn Deutsch reden hören. Typically such constructions are created by a "zu + inf", like
"I habe ihn Deutsch zu reden gehört, but "sehen", "hören" are exceptions. They work like a modalverb and we have no "zu" with them.
Here we have another surprising effect: in fact, "hören" und "sehen" are so modalverb-like, that they don't get even a perfekt-form, if we use them as modalverb! Thus, the correct sentence is:
Ich habe ihn Deutsch reden hören.
There are around 10 verbs having this feature. If you miss them, native speakers will catch it on the spot, but they will understand your sentence without any problem.