All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
We advise you first to stay away from exaggeration and whispers.
Simply using the names Muhammad and Ibrahim and the like, which are not intended for prophets, does not require reverence.
As for the names intended for the prophets, they must be revered, and disrespecting them is forbidden, like other revered things in Sharee'ah, such as the names of Allah the Almighty or the verses of the Quran, even if written in a language other than Arabic.
It is obligatory to pick them up if found in a dirty place. However, this obligation is restricted to what does not lead to undue hardship and unusual difficulty; otherwise, it is not obligatory. Allah, the Exalted, says (what means): {And He has not placed upon you in the religion any difficulty.} [Quran 22:78]
It is not obligatory to inspect garbage and the like in search of discarded papers containing religiously revered names; this would be excessive and burdensome.
Allah knows best.