Have you tried exporting your component from the shared module? If so, you can then use component outputs (if they are adjacent) or a service (if they are not adjacent) like you mentioned to share data between two components.
import { NgModule } from '@angular/core';
import { CommonModule } from '@angular/common';
import { MyComponent } from '../components/my-component';
@NgModule({
imports: [
CommonModule,
],
declarations: [
MyComponent,
],
exports: [
MyComponent,
],
})
export class SharedModule { }