1. Individual service companies / Sole proprietorship
One thing to be aware of is that in many European companies a person who works for you full-time is considered to be an 'employee' even if they come in through an individual service company. In particular if they have no other active 'customers'.
The more you treat people as 'employees' the more at risk you are: if you give them holidays, if you monitor their work closely etc, if they don't invoice you, etc.
The risk is that you will have to pay employment taxes and benefits and you may be fined.
Pragmatically, you are unlikely to be at risk at a low number of team members within a jurisdiction: say up to 10-15. But, at some point you will be considered to have created a permanent establishment from a tax position.
2. PE vs Limited Company
Many people talk about establishing a company in a jurisdiction and then employing people etc. There's actually a state before this where you have a 'permanent establishment' from a tax position even if you don't have a limited company in the jurisdiction. You can be considered to have employees at this point and you are responsible for payroll taxes, but don't yet incur the business taxes (e.g. corporation tax). Unfortunately, it varies by European jurisdiction how likely this is.
This state is essentially supposed to handle the idea of someone having an employment contract in one jurisdiction (say France), while living in another (say Germany). Practically, it doesn't work yet because the rest of the employment rights assume that you have a company in each jurisdiction. You can sort of hack it for an intermediate step.
3. Pragmatically expanding in Europe
How you handle this is up to you in terms of how comfortable you are with the risk profile. The elements are too long (and boring) to go into here.
In my personal opinion it makes sense to:
1. Hire team members in a limited number of jurisdictions.
Basically where you believe you will eventually establish a full company: you might pick three jurisdictions you're comfortable with.
2. Handle team members as employees: this is a basic decision about how you want to handle your remote team but it has big impacts. If you treat them as 'employees' then make it clear to them that they are responsible for their employment tax, pensions, etc.
3. Risk the permanent establishment tax problem
4. When a jurisdiction gets to 10 or so people form a company. In many European countries this is around the stage where employment rights also kick in.
5. Reverse the people into the company and pay the employment taxes/benefits etc.
This will cause problems because you will have to adjust pay since you will now be incurring payroll taxes and public pensions etc.