If I commit a sin like mockery or backbiting in front of someone, must I inform them of the act's prohibition, or should I rely on their prior knowledge of it? And would my failure to inform them subject me to the warning in the hadith: "Whoever institutes a bad practice will bear its sin and the sin of whoever acts upon it until the Day of Resurrection"?

It is not a condition for the validity of repentance to inform those who witnessed the sin that the act is forbidden. However, teaching people good and warning them against evil is a highly emphasized religious matter. There is a concern that whoever was emulated by someone in a sin might bear its burden, due to the Prophet's (peace be upon him) saying: "Whoever introduces a bad practice in Islam will bear its sin and the sin of whoever acts upon it after him." If you repent with sincere repentance, we hope that Allah will forgive you and not hold you accountable for the sins of those who imitated you, for Allah is Forgiving and Merciful and accepts sincere repentance no matter how great the sin. Scholars have stipulated that the repentance of one who repents from a sin is accepted, and the lingering effect of their sin does not harm them.