There's no problem in using an Azure Service from one Resource Group in another. Resource Groups do not separate resources, they're just a grouping.
A container that holds related resources for an Azure solution. The resource group can include all the resources for the solution, or only those resources that you want to manage as a group. You decide how you want to allocate resources to resource groups based on what makes the most sense for your organization.
The text below was taken from What is an Azure resource group?
A resource group is simply an identifier that Azure Resource Manager applies to resources to group them together. This resource group ID allows Azure Resource Manager to perform operations on a group of resources that share this ID.
A resource group cannot include resources from different subscriptions
And last but not least, take a look at this Resource Groups explanation. Especially look at item 1 in the list:
All the resources in your group should share the same lifecycle. You deploy, update, and delete them together. If one resource, such as a database server, needs to exist on a different deployment cycle it should be in another resource group.