it is hard to answer your question because there is very little information on details.
We are talking about employees here, so one has to keep in mind that there is a legal framework, an employment contract. So especially in small companies and startups trusting the employees is the most practical solution. How do you manage security-related OCD may be an interesting read.
It probably is a good starting point to define one person (perhaps with substitute if your startup is large enough) to be responsible for IT. It is common practice that the IT guys are trusted in small companies because this get impractical otherwise very quickly.
Then you should look into the concrete services which require authentication.
Let me give an examples:
You probably have an email@example.com email address that several people shall be able to access. Setting this up as a normal account, with a password shared between the employees responsible for it, is an easy solution. Simple solutions are good because they are easily understood.
If one of those employees leaves the company, it has to be changed. Another downside is that it is an other password people have to remember. In case of an incident you will have to fallback to ip-addresses when reading the log files because all employees in question share one account.
Many cooperate mail servers such as Microsoft Exchange allow sharing of postboxes with specific accounts. So the employees responsible for the contact-email-address can access it using their own account. This is obviously a preferable solution if it is offered by your email system.
Another example are access to servers. On Microsoft Windows servers it may be a good idea to set up a domain. This allows people to login with their domain account to various Microsoft based computers and you can define roles for each account/computer pair.
The problem still exists for the domain controller itself. But this one should be operated only by two people, so preventing the issues of complex combinations between people and servers.
For the administration of Unix servers sudo (man sudoers) is the equivalent because it allows people to execute commands with the permissions of another user.
To summarize: Trusting your employees is a practical solution. There are relatively simple solutions for all common services but each service needs to be looked at.