They are not sentence fragments. Each sentence consists of two clauses connected by the conjunction but. In such a case, unless the subject if the second clause is different, then it is automatically assumed to be the same as that of the first. (Notice how the same applies to “that of the first” in the previous sentence: the structure, with “of the second clause, followed by “of the first, requires that the ‘missing’ noun is understood to be clause.
In your examples, you can either leave them as they are, or use a definite article (he/she/it etc.) or repeat Peter, etc. But would it be objectionably repetitive?
In your examples, no. Repetition is a matter of style, not of grammar. Some repetition is clumsy or tiresome:
My trousers are brown, and my trousers are tight, and my trousers fit me perfectly.
This is a real clunker. It says repetitively what could just as well be expressed as:-
My trousers are brown, tight and fit me perfectly.
But the faults are stylistic, though the repetition of ‘and’ is a breach of convention.
Not only does repetition have nothing to do with grammar, it can be rhetorically and poetically effective.
Alone, alone, all, all alone,
Alone on a wide wide sea!
And never a saint took pity on
My soul in agony. (Coleridge Rhyme of the Ancient Mariner)
This Senate is tired of your lying, tired of your deceit but above all this Senate is tired of your grossly unjustified arrogance. (made up as far as I know)
These repetitions reinforce a picture, a feeling or an image. They are a matter of taste, except to say that Coleridge’s is sublime and mine is anything but.