[Disclosure: I work for CoSign. We make digital signature products.]
In response to the original question and the comments from sk1 to another answer:
Standard digital signatures or "digital signature"?
I think that the question is about standard digital signatures, eg for PDFs the most real-world, harmonized document is PAdES. See the Standards tab of the ETSI page. This standard is harmonized with the Digital Signatures section of the PDF specification, the NIST digital signatures standard, etc.
Verification
Assuming the above, the answer about verifying the signed documents is that there are many software apps that can verify standard digital signatures. This can be done at people's workstations, on a server, via a browser, mobile app, etc. Eg:
- On a PC, Adobe Reader (or Acrobat) can verify the digital signatures on a PDF file. So can Nitro Pro, Bluebeam Revu, Foxit Phantom, and other PDF software.
- On a PC, Microsoft Word and Excel have native features for digitally signing Word/Excel documents and for verifying signatures on signed documents. Note that signing requires a signing infrastructure that supports the local APIs that the apps use: CAPI / CAPI NG
- On a server, you can sign and verify PDF, Word, and Excel documents by using software/hardware from different vendors. The software is often integrated into an ECM system such as SharePoint. My company, CoSign, has mature SharePoint Server and SharePoint Online digital signing products.
- For browser access, you need a web app that supports signing and verification. Eg CoSign Web App. By the way, some web apps, including CoSign, can sign and verify Word and Excel documents, in addition to the usual PDF files.
- Digital signature apps are also available for iOS and Android. Due to the security issues of the private key, these apps are clients to the actual signing service where the private key is stored.
Private key storage
The question also touched on the issue of storing the digital certificate / private keys of the signers. Industry norms and standard practices require that the signers' private keys be stored in a Secure Signature Creation Device (SSCD). Originally, the idea was that edge devices (Smart Cards) be used as SSCDs.
But as we all know, edge devices have many practical issues of deployment, maintenance, access, etc. So now, centralized SSCD server appliances are available and recognized by the standards bodies. An interview about these issues.
The centralized SSCD appliance sits within your network and holds the signers' private keys, certs and other data. It is a hardened box with anti-tampering technologies that render the private keys useless if the box is opened. It should be certified by a recognized independent testing laboratory. CoSign and other vendors make these appliances.
Return on Investment (ROI) is usually a year or less.