Only allow original uploader or site admin to delete attachments.

attachments.php

<?php if ( bp_docs_is_doc_edit() || bp_docs_is_doc_create() ) : ?>
	<?php bp_docs_media_buttons( 'doc_content' ) ?>
<?php endif; ?>

<?php 
// Maybe you'll want to do something with main doc ID.
$doc_id = get_the_ID(); 
?>

<ul id="doc-attachments-ul">
<?php foreach ( bp_docs_get_doc_attachments() as $attachment ) : ?>
	<?php
	$markup = '';

	$att_url    = bp_docs_get_attachment_url( $attachment->ID );

	$att_base   = basename( get_attached_file( $attachment->ID ) );
	$doc_url    = bp_docs_get_doc_link( $attachment->post_parent );

	$attachment_ext = preg_replace( '/^.+?\.([^.]+)$/', '$1', $att_url );

	$attachment_delete_html = '';
	// Only allow the original uploader to delete that attachment.
	if ( ( current_user_can( 'bp_moderate' ) || bp_loggedin_user_id() == $attachment->post_author ) && ( bp_docs_is_doc_edit() || bp_docs_is_doc_create() ) ) {
		$attachment_delete_url = wp_nonce_url( $doc_url, 'bp_docs_delete_attachment_' . $attachment->ID );
		$attachment_delete_url = add_query_arg( array(
			'delete_attachment' => $attachment->ID,
		), $attachment_delete_url );
		$attachment_delete_html = sprintf(
			'<a href="%s" class="doc-attachment-delete confirm button">%s</a> ',
			$attachment_delete_url,
			__( 'Delete', 'buddypress' )
		);
	}

	$markup = sprintf(
		'<li id="doc-attachment-%d"><span class="doc-attachment-mime-icon doc-attachment-mime-%s"></span><a href="%s" title="%s">%s</a>%s</li>',
		$attachment->ID,
		$attachment_ext,
		$att_url,
		esc_attr( $att_base ),
		esc_html( $att_base ),
		$attachment_delete_html
	);

	echo $markup;
	?>
<?php endforeach; ?>
</ul>