Information derived from Étienne's Comment
In summary either the employer directly submits the social insurance to the relevant Austrian authorities or the employer hands over all the money, gross salary plus employer social insurance contributions, to the employee who in turn submits the amount themselves to the relevant Austrian authorities. The below link ( in German details the process )
https://media.arbeiterkammer.at/tirol/Wirtschaft/MB_Arbeiten_ausl_AG_2015.pdf
Specifically Part 2 in the linked document is most interesting, translated into English.
Employers from EU/EEA countries or Switzerland:
In this case, the
employer is in principle obliged to provide both the employee as well
as the employer's social security contributions to TGKK, and to pay
the remaining amount (gross minus SI) to the employee. However, there
is a possibility that the employer and the employee may not be in
agreement concludes a written agreement (according to Article 21
paragraph 2 of Regulation 987/09), according to which the employee
pays the social security contributions himself/herself and in return,
the employer undertakes to provide the employee, in addition to the
employer' s gross salary, the employer's contribution. The employee is
responsible for then the employer's contribution (21.7% for workers
and 21.83% for employees) and its own contribution (18.2% for workers
and 18.07% for salaried employees) to the TGKK. These contribution
rates are valid from an income of € 1,497 gross to to the maximum
contribution basis (€ 4,440. -). The exact contribution amounts can be
found on the homepage of the main association of social insurance
companies. If the employee concludes a written agreement with the
employer over the payment of gross wages plus employer contributions,
he/she can easily verify whether he/she is covered by social security,
as he/she has it in his/her own hands, checking the social security
contributions to be discharged. In the event of non-payment of
contributions to the employee, the employee must themselves take
action against the foreign employer. In addition to this written
agreement, form (EUVereinbarung nach Art. 21 VO 987/2009) should be
submitted. It can be found on the TGKK homepage.
Translated with DeepL
