so what's the trouble? make a first script also singletone, make there static method to receive new text, and call it whenever you need.

if it can't be a singletone, declare public variable of type 'TextMeshExample' in 2nd script

public TextMeshExample textMeshExample;

assign GameObject with TextMeshExample script attached to GameObject that keeps 2nd script, for it you need save script, go to inspector and drag-n-drop GO with 1st script to GO with 2nd script, exactly to variable 'textMeshExample' we just made.

now, add a method to TextMeshExample that will let us set custom text there:

public void SetCustomText(string text)
{
    textMesh.text = text;
    textMesh.Commit();
}

and the last, call from second script the first one. (in SetDistance method?)

textMeshExample.SetCustomText("my text");

first script - result

using UnityEngine;
using System.Collections;

public class TextMeshExample : MonoBehaviour
{
	tk2dTextMesh textMesh;
	int score = 0;

	// Use this for initialization
	void Start()
	{
		textMesh = GetComponent();
	}
	void Coin()
	{
		score += 1;
		textMesh.text = " " +score.ToString();
		textMesh.Commit();
	}
	public void SetCustomText(string text)
	{
		textMesh.text = text;
		textMesh.Commit();
	}
}

second script - result

using UnityEngine;

public class GUIManager : MonoBehaviour
{
	public GUIText distanceText;
	public TextMeshExample textMeshExample;
	private static GUIManager instance;
	void Start()
	{
		instance = this;
		textMeshExample.SetCustomText("my text");
	}
	void Update()
	{
		GameEventManager.TriggerGameStart();
	}

	public static void SetDistance(float distance)
	{
		instance.distanceText.text = distance.ToString("f0");
	}
}