Is it usually the tenant or the landlord who pays for wastewater, i.e., water out?

It depends on whether you supply cold water to the tenant and whether you have set up an advance payment arrangement for it.

If you supply cold water, you can, according to § 68, require your expenses for the tenant's consumption to be reimbursed — and according to § 72, the water account can include all expenses for water supply, including water and wastewater charges (i.e., wastewater). However, this requires that you have notified the tenant of advance contributions for water with at least 6 weeks' notice according to § 71, and that a consumption account is continuously maintained according to §§ 72-73.

If such an arrangement is not set up, and wastewater is not specifically mentioned in the lease agreement, the expense is usually included in the rent, and then it is you as the landlord who bears it.

Do you already have an advance payment arrangement running for water, or is there something specific about wastewater in the lease agreement? This determines whether you can charge it separately to the tenant.