STSM015010 - Introduction to Stamp Duty on shares and Stamp Duty Reserve Tax (SDRT): Stamp duty - payment and penalties: who pays Stamp Duty?
The legislation governing Stamp Duty does not say who is responsible for paying it. What it does do is to say that an instrument is not available for any purpose whatsoever other than as evidence in criminal proceedings (section 14 (4) Stamp Act 1891) unless it has been duly stamped (see STSM141030).
In practice, this normally results in an instrument that needs to be stamped being presented for stamping by the person who wishes to use it. For example, a stock transfer form would normally be presented by the purchaser as he would wish to use it to register his ownership of the shares in question.
Likewise, an SH03 form used to give notice of a purchase by a company of its own shares would be presented by the company to HMRC for stamping so that it could then register the form at Companies House.
In both of the examples given above the person required to act upon the document (the company registrar and Companies House respectively) will refuse to do so unless the document is stamped appropriately. Further information can be found at STSM151010 and STSM151020.