The share transactions of private companies are just that – private. They do not have to be declared to HMRC, unless a taxable capital gain or loss arises.
However, company law requires the company to record at Companies House the names of people who hold 25% or more of the votes or shares in the company. These are referred to as ‘people with significant control’ (PSC).

If the PSC change, HMRC will assume that some shares have been sold or transferred and that someone has made a capital gain in the process.
HMRC has cross-checked the Companies House records of changes in PSC for private companies with the tax returns of those people who appear to have lost control. In such cases HMRC has written to the individuals to ask why a gain/loss was not declared on their 2019/20 tax return.

There may be no gain to declare, where the taxpayer died or made a share for share exchange. Alternatively, a small gain may have arisen, which does not need to be declared if the total gains for the year are less than the annual exemption (£12,000 for 2019/20).