This depends on whether in this society there are institutions that seek to benefit from taking on someone of status

The United States, nor the Soviet Union had any lack of celebrities and/or national heroes. Whether profit or propaganda, if a name has recognition, it has the potential to sway public opinion and no matter the direction of your society you will always have those able and willing to wield that. However in the case that the state has no care, and/or there are no corporations to profit, the concept of "fame" would be a highly subdued version of the one we associate with the word today.