The Straßenverkehrsordnung(Translation) in Germany says this:

§ 34 Accidents

(1) Any person involved in a traffic accident must:

1. stop immediately;

2. take the necessary steps to ensure traffic safety and, if there is only minor damage, move their vehicle to the edge of the carriageway;

3. ascertain the consequences of the accident;

4. assist injured persons (section 323c of the Penal Code);

5. to other persons present at the scene of the accident who were involved and have suffered damage:

   a) state that they (i.e. the person referred to in the first clause) were involved in the accident; and

   b) if requested to do so, provide their own name and address, present their own driving licence and vehicle registration document and, to the best of their knowledge, provide details of their third-party insurance;

6. a) remain at the scene of the accident until, by virtue of their own presence, it has been possible to identify their personal details, their vehicle and the nature of their involvement to the benefit of the other persons who were involved in and have suffered damage in the accident; or

   b) wait for a reasonable length of time and leave their own name and address at the scene of the accident if nobody was prepared to perform the identification;

7. immediately facilitate subsequent identification if they have left the scene of the accident legitimately, after giving a plausible excuse or upon expiration of the waiting time (paragraph 6(b)). For this purpose, they must inform at least the persons referred to above (paragraph 6(a)) or a nearby police station that they were involved in the accident and must provide their own address, their present whereabouts as well as the registration number and location of the vehicle that was involved in the accident, which must be kept available for immediate investigation for a reasonable length of time.

For minor accidents (no injuries, no suspected criminal offence, no major disagreements between involved parties, ...) you do not have to call and wait for police (you have a right to call them but they might try to discourage you from requesting that they come to the scene). For normal traffic accidents insurance doesn't come and collect evidence at the scene, instead an appraiser will document damage to your car and look at statements and witness reports and other documentation. You can just exchange all necessary information with the other party/ies and then leave with your car (possibly using a towing service).

The Straßenverkehrsordnung(Translation) in Germany says this:

§ 34 Accidents

(1) Any person involved in a traffic accident must:

1. stop immediately;

2. take the necessary steps to ensure traffic safety and, if there is only minor damage, move their vehicle to the edge of the carriageway;

3. ascertain the consequences of the accident;

4. assist injured persons (section 323c of the Penal Code);

5. to other persons present at the scene of the accident who were involved and have suffered damage:

   a) state that they (i.e. the person referred to in the first clause) were involved in the accident; and

   b) if requested to do so, provide their own name and address, present their own driving licence and vehicle registration document and, to the best of their knowledge, provide details of their third-party insurance;

6. a) remain at the scene of the accident until, by virtue of their own presence, it has been possible to identify their personal details, their vehicle and the nature of their involvement to the benefit of the other persons who were involved in and have suffered damage in the accident; or

   b) wait for a reasonable length of time and leave their own name and address at the scene of the accident if nobody was prepared to perform the identification;

7. immediately facilitate subsequent identification if they have left the scene of the accident legitimately, after giving a plausible excuse or upon expiration of the waiting time (paragraph 6(b)). For this purpose, they must inform at least the persons referred to above (paragraph 6(a)) or a nearby police station that they were involved in the accident and must provide their own address, their present whereabouts as well as the registration number and location of the vehicle that was involved in the accident, which must be kept available for immediate investigation for a reasonable length of time.

For minor accidents (no injuries, no suspected criminal offence, no major disagreements between involved parties, ...) you do not have to call and wait for police (you have a right to call them but they might try to discourage you from requesting that they come to the scene). For normal traffic accidents insurance doesn't come and collect evidence at the scene, instead an appraiser will document damage to your car and look at statements and witness reports and other documentation. You can just exchange all necessary information with the other party/ies and then leave with your car (possibly using a towing service).

However, OP might refer to the specific case of an accident with a rented car. In that case, the contract with the rental service often mandates that you call police and request an official police documentation and report in case of an accident.