It varies by office and by firm. I can speak for McKinsey USA and UK - you can train to be an interviewer for round 1 interviews after a year of being at the Firm (for Associates and above) and two years for Business Analysts. You can only interview for entry at your level or below. So Associates who have been an Associate for at least a year can start interviewing for Associate level and below, BAs who have been at the Firm for at least two years can interview for BAs. 
 

Final round is always done by Partners and sometimes Associate Partners. 

The only consistent rule is you need more than a year of tenure at McK and have received interviewer training.  Based on my experience, there was no connection with consultant performance.  The role of your interviewer has a large range.

1st round (some offices like the US have started with one power round): Senior BA to Partner including recruiters (typically Senior BA, Associate with more than a year of tenure, EM)

Final round: EM to Senior Partner including Head of Recruiting (most common is AP and Partners)

(editiert)

Hi there,

Depends on the office but what some others have said is just plain wrong. 

You can become an interviewer as early as 1-year tenure within the firm (far below EM), if you have performed well.

Cheers,

Florian

Hi there, 

for the 1st round of interview you can be interviewed by senior level consultant (equivalent BCG). From 2nd round only Project leader and up. 

All the best,

Lucie

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Hello,

Generally you would start being allowed to interview at the Associate (McK)/Consultant (BB) roles for first rounds. Second rounds are done by EM/project leader and above.

Dear Anonymous A,

I shall speak from my McKinsey experience.

At McKinsey, Associates rarely conduct First Round interviews.  It's probably more accurate to say they never do, but I cannot be certain. 
Typically, Associates who have successfully transitioned to the Engagement Management role and continue to be rated highly (the top tier that includes the roughly top 20% of performers within the same hierarchy) are tapped to contribute to the firm's talent acquisition by becoming interviewers.

Once an EM is tapped (and ultimately accepts to interview…I never met a single EM who said No…McKinsey is not a place where people generally say No in such circumstances), they go through a training course, are given access to the rotation of interview Cases for that office, and then assigned to support Recruiting.

I have heard of smaller offices where EMs outside the top-performing tier were also tapped, but this likely would have been due to a sponsoring partner with clout vouching for the EM's ability to serve office and firm well in this regard.

I can't see analysts (senior or not) being tapped to conduct interviews!