diff --git a/guppylang/module.py b/guppylang/module.py
index d7d67fec..11c96400 100644
--- a/guppylang/module.py
+++ b/guppylang/module.py
@@ -113,7 +113,6 @@ def load(
                 module = imp.id.module
                 assert module is not None
                 module.check()
-                defs[imp.id] = module._checked_defs[imp.id]
                 names[alias or imp.name] = imp.id
                 modules.add(module)
             elif isinstance(imp, GuppyModule):
@@ -123,7 +122,6 @@ def load(
                 defn = ModuleDef(def_id, name, None, imp._globals)
                 defs[def_id] = defn
                 names[name] = def_id
-                defs |= imp._checked_defs
                 modules.add(imp)
             elif isinstance(imp, ModuleType):
                 mod = find_guppy_module_in_py_module(imp)
@@ -135,16 +133,15 @@ def load(
         # Also include any impls that are defined by the imported modules
         impls: dict[DefId, dict[str, DefId]] = {}
         for module in modules:
+            # We need to include everything defined in the module, including stuff that
+            # is not directly imported, in order to lower everything into a single Hugr
+            defs |= module._imported_checked_defs
+            defs |= module._checked_defs
             # We also need to include any impls that are transitively imported
             all_globals = module._imported_globals | module._globals
-            all_checked_defs = module._imported_checked_defs | module._checked_defs
             for def_id in all_globals.impls:
                 impls.setdefault(def_id, {})
                 impls[def_id] |= all_globals.impls[def_id]
-                defs |= {
-                    def_id: all_checked_defs[def_id]
-                    for def_id in all_globals.impls[def_id].values()
-                }
         self._imported_globals |= Globals(dict(defs), names, impls, {})
         self._imported_checked_defs |= defs
 
diff --git a/tests/integration/test_imports.py b/tests/integration/test_imports.py
index 49443740..7e98f442 100644
--- a/tests/integration/test_imports.py
+++ b/tests/integration/test_imports.py
@@ -177,3 +177,28 @@ def test(x: int) -> int:
         return implicit_mod.foo(x)
 
     validate(module.compile())
+
+
+def test_private_func(validate):
+    # First, define a module with a public function
+    # that calls an internal one
+    internal_module = GuppyModule("test_internal")
+
+    @guppy(internal_module)
+    def _internal(x: int) -> int:
+        return x
+
+    @guppy(internal_module)
+    def g(x: int) -> int:
+        return _internal(x)
+
+    # The test module
+    module = GuppyModule("test")
+    module.load_all(internal_module)
+
+    @guppy(module)
+    def f(x: int) -> int:
+        return g(x)
+
+    hugr = module.compile()
+    validate(hugr)