Contracts between the original record custodian and FamilySearch govern nearly all of the records within the collections on FamilySearch.
For most contracts, FamilySearch acquires rights for you to use the records merely for incidental, personal, noncommercial genealogical research purposes. Your rights include the ability to extract factual data about your direct family line. You can put that data into your personal family tree.
Contracts prohibit the publication or distribution of the actual record images or documents (including through print or the web). Contracts also prohibit wholesale indexing, transcribing, or translating of the records, even when these activities are for nonprofit purposes. You must have written permission from the custodian of the original records before you publish an image of a record or document.
From the collection description screen, click How to Use this Collection. The system routes you to a wiki article about the collection. Some of these wiki articles indicate who the custodian of the collection is.
FamilySearch Terms of Use provides additional details. In addition, you can find contact information to request permission to use materials.