Your whole project should build straight from very fresh boxes, and doing builds on developers machines will never be fresh boxes. It's hard to remove the cult knowledge from a development team. My project just discovered a new unlisted dependency when doing a deploy, because every developer knew about it and installed it on their machine beforehand. It was explicitly listed in the build/devtools dependencies, but not supposed to be a runtime dep. Had the developers run tests on a fresh machine, they'd have run into it. (Of course, the CI team had also installed it on the CI boxes, because they used it for debugging.)
For a large class of tests, having the developers run them in the build environment is fine. But you also need to run them in the deploy environment, and to that end developers should hit the CI system. Every test that the CI system runs on each commit should be runnable before the commit. You should have tooling and spare capacity such that the CI system is used to run tests immediately before commit, not right after - That's too late. You should run them whenever a dev sends off for code review. You should run whenever a dev feels like it; If they're in a good spot, run the unit tests, run the CI tests, see what's broken.