What do we need to take into account where valuing a private company?

Tons of stuff but let me touch only on 1:-

The talents running the company.

The founders, the current operating teams. These are the real gems of the company.

There are many instances where big companies swallowed a start up for the sake of the talents.

And many times the startup will fail if the talents leave.

Always look at the company and analyse, will the company still be as valuable without the core team now?

It depends on the stage at which the private company is at as well.

Coming from a Venture Capitalist perspective, a very simple thought flow is as such:

Let's say that an early stage Software-as-a-Service is raising at 5x price to sales. You look at comparables and see that other similar companies have raised at 3x P/S. Based on this, you then need to evaluate other factors such as quality of team (are they serial entrepreneurs who have exited successfully before?), Technical due-diligence of their product (is it really superior?), the veracity of their go-to-market strategy.

It is also common to tri-age this with other valuation mehods, but if you are looking at a super early stage startup, it could be challenging as financial historicals are limited.