Do threads have a distinct heap?

By default, C has only a single heap.

That said, some allocators that are thread aware will partition the heap so that each thread has it's own area to allocate from. The idea is that this should make the heap scale better.

One example of such a heap is Hoard.

No. All threads share a common heap.

Each thread has a private stack, which it can quickly add and remove items from. This makes stack based memory fast, but if you use too much stack memory, as occurs in infinite recursion, you will get a stack overflow.

Since all threads share the same heap, access to the allocator/deallocator must be synchronized. There are various methods and libraries for avoiding allocator contention.

Some languages allow you to create private pools of memory, or individual heaps, which you can assign to a single thread.