If you're only having sex with your partners and your partners are only having sex with you and each other, is it still important for you and your partners to get tested regularly?
If the group is entirely closed - if Anna only has sex with Ben and Carter and Dana, and Ben only has sex with Anna and Carter and Dana, and Carter only has sex with Anna and Ben and Dana, and Dana only has sex with Ben and Anna and Carter - then no, regular testing is less critical.
Everyone should be tested yearly as part of an annual checkup (or talk to their sexual healthcare provider about what’s best for them), and everyone should have up-to-date test results at the beginning of the relationship. As long as everyone is on the same page with regards to safety, boundaries, and the nature of the relationship, it functions STI-risk-wise the same as a monogamous relationship between two people.
But if things change in any way - if Carter has sex with Samantha - then everyone needs to be tested regularly.