Processing all passed overloads at once

I'm tired of making up on the spot debug codes and including <iostream> in every single file. So I wanted to make myself a universal, self-contained and lightweight debug class, that I would just include in the header, and forget.

I want to use something along the lines of

#include "debug.hpp"
debug DBG;
DBG << "foo and" << " bar";
//Or even better, just include it and do debug() << "foo and" << " bar";

So, I wrote this:

#include <iostream>
#include <string>
#include <chrono>
#include <ctime>

class Debug
{
public:    
  Debug &operator<<(std::string arg_0)
  {
    auto tempTime = std::chrono::system_clock::to_time_t(
      std::chrono::system_clock::now() );
    auto timeString(ctime(&tempTime));
    timeString = timeString.substr(timeString.find(':') - 2, 8);

    std::cout << timeString << " >> " << arg_0 << '\n';
    return *this;
  }
};

But of course, this doesn't work because, as I've learned, every overload operator causes this function (is it still called a function?) to trigger separately. Creating:

hour:minute:second >> foo and
hour:minute:second >> bar

Any way I could pass everything at once after the first overload operator appears? Maybe as a stringstream? Also, I won't be only passing strings, but anything that I need, will this require me to manually create a separate overload function for every signle type that I may pass?

P.S: Cross-plaform solution is optional, but welcome (Currently developing on Linux)