Removing the __MODULE__. from othermacro fixes the require issue, but exposes a bigger problem:

defmodule Test do
  defmacro somemacro(arg1, do: block) do
    quote do
      defmodule unquote(arg1) do 
        def unquote(arg1), do: 1
        unquote_splicing(elem(block, 2))
      end
    end
  end

  defmacro othermacro(arg2, arg1, do: block) do
    newblock = quote do
      def unquote(arg2), do: 2
    end

    somemacro arg1, do: newblock
  end
end

which fails with:

** (ArgumentError) expected a list with quoted expressions in unquote_splicing/1, got: nil
    (elixir 1.11.2) src/elixir_quote.erl:185: :elixir_quote.argument_error/1
    (elixir 1.11.2) src/elixir_quote.erl:166: :elixir_quote.list/2

This happens because when othermacro calls somemacro, the value somemacro sees in block is {:newblock, [line: 17], nil}.

One way to avoid this problem is to keep the defmacro parts simple and depend on plain functions to generate the actual AST fragments:

defmodule Test do
  defmacro somemacro(arg1, do: block) do
    # wrapped block in a list because unquote_splicing expects a list of quoted expressions
    build_ast(arg1, [block])
  end

  defmacro othermacro(arg2, arg1, do: block) do
    newblock = quote do
      def unquote(arg2)(), do: 2
    end

    # the old code didn't use block (?) 
    build_ast(arg1, [block, newblock])
  end

  defp build_ast(arg1, block) do
    quote do
      defmodule unquote(arg1) do 
        def unquote(arg1)(), do: 1
        unquote_splicing(block)
      end
    end
  end
end

defmodule FooMod do
  require Test

  Test.othermacro(:foo, :bar, do: "not used")
end

BUT BEWARE SOME GOTCHAS

The module defined by Test.somemacro is not nested - so in the example, it generates a top-level module named :bar:

iex(26)> :"Elixir.FooMod.bar".foo
** (UndefinedFunctionError) function :"Elixir.FooMod.bar".foo/0 is undefined (module :"Elixir.FooMod.bar" is not available)
    :"Elixir.FooMod.bar".foo()

iex(26)> :bar.foo                
2

This may not be what you want.