The LSP server performs no network access in itself, but runs cargo metadata which will update or download the crate registry and the source code of the project dependencies.
The Code extension connects to GitHub to download updated LSP binaries and, if the nightly channel is selected, to perform update checks.
Any other editor plugins that integrate with rust-analyzer are not under the control of the rust-analyzer developers. For any privacy concerns, you should check with their respective developers.
If cargo check is enabled (the default), any build scripts or procedural macros used by the project or its dependencies will be executed. This is also the case when cargo check is disabled, but build script or procedural macro support is enabled in rust-analyzer (off by default).